Kinematics

This section explores key concepts of motion, including projectile motion, friction, circular motion, banking, and skidding. Projectile motion involves the curved paths objects follow under the influence of gravity. Friction, the resistive force between surfaces in contact, impacts movement and stability. Circular motion examines objects moving in curved paths, analysing the forces required for such motion. Banking refers to the tilting of roads or tracks to help vehicles navigate curves safely, while skidding highlights the loss of traction, often resulting from inadequate friction. These areas connect theory with real-life applications, such as sports, transportation, and engineering, fostering an understanding of forces and motion in various scenarios. By analysing these principles, key problem-solving skills can be developed, enhancing comprehension of both natural and man-made systems.

KEY IDEAS

• A projectile is any object that is launched or thrown into the air and moves solely under the influence of gravity

• Banking is the act of constructing roads in a manner that will help cars negotiate curves at relatively higher speeds.

• Centripetal force is the force that is required to keep bodies moving in a circular track.

• Circular motion is a type of motion in which an object moves around a fixed point while maintaining a constant distance from that point.

• Friction is the force that opposes the relative motion or the tendency of such motion of two surfaces in contact.

A projectile is any object that is launched or thrown into the air and moves solely under the influence of gravity (i.e. all other forces are negligible). Common examples include kicking a football, throwing a javelin, or tossing a handball into the air.

Figure 5.1: Image of an athlete throwing a javelin When a projectile travels through the air without any additional propulsion, it follows a curved path known as a parabolic trajectory. This motion is referred to as projectile motion. While forces like air resistance may act on the projectile, their effects are typically minor compared to gravity and are often ignored in simple analyses of projectile motion.

Components of projectile motion Have you ever tried plucking a mango fruit from a tree using a stone before? What can you say about the direction of motion of the stone? It travels both upward (vertically) and forward (horizontally) at the same time. This means both initial velocity and displacement of stone have components in the vertical and horizontal directions. Let’s consider the details of these components:

Figure 5.2: a diagrammatic representation of the projectile motion of a football Equations of motion to remember: v = u + at, v²= u²+ 2as, s = ut + 1__ 2at²In projectile motion, we will use y or sometimes h in place of s for a vertical distance, x in place of s for a horizontal distance and g in place of a.

Given the angle of projection θ, the initial velocity u has the vector components uₓ = ucosθ and u_(y) = usinθ in the horizontal and vertical directions respectively.

The final velocities are accordingly given as vₓ and v_(y).

Activity 5.1 Resolving a velocity into its horizontal and vertical components Study the worked example below carefully before attempting the example questions that follow.

Worked Example

A soccer ball is kicked with an initial velocity of 20 m/s at an angle of 30 degrees above the horizontal. Draw a diagram and calculate the horizontal and vertical components of the initial velocity.

Solution

Remember:

The horizontal component is calculated using the formula:Vₓ = V cos(θ) The vertical component is calculated using the formula: V_(y) = Vsin(θ) Horizontal Component Vₓ = Vcos(θ) = 20cos(30°) = 17.32 m / s Vertical Component V_(y) = V sin(θ) = 20 sin(30°) = 10 m / s Practice questions Now, using the worked example as a guide, solve the following problems individually or in groups.

1. A plane flies at a constant speed of 500 km/h at an angle of 25° north of east.

Draw a diagram and calculate the horizontal and vertical components of the velocity.

2. A diver leaps off a cliff with an initial velocity of 8 m/s at an angle of 45° below the horizontal. Draw a diagram and calculate the horizontal and vertical components of the velocity.

The horizontal component Since acceleration due to gravity does not act in a horizontal direction, g = 0 for horizontal motion.

Therefore, substituting in to v = u + at , v = ucosθ + (0)t = ucosθ .

This means that the velocity in the horizontal direction does not change in the course of the motion.

Also, substituting in to s = ut + 1__ 2at², x = utcosθ + 1/2(0) t²= ut cosθ Summary for horizontal motion:

vₓ = ucosθ, x = utcosθ The vertical component The vertical component of the motion is either upward or downward, hence g is either (+) or (-). The following equations will then be applied accordingly.

v_(y) = usinθ ± gt v_(y) ²= u²sin²θ ± 2gy y = utsinθ ± 1/2 g t²Aside from the quantities considered above, you may ask, how long could a projectile take to land? How high could the projectile go before beginning a downward motion? How far could it land from the point of projection? etc. See Annex B for an analysis of how to find these values, but remember that all of the questions above can be solved by accurate use of the three equations of motion.

Activity 5.2 Exploring Projectiles in Sports

Objective: Work in pairs to identify different sports and the objects that act as projectiles in those sports. You will discuss the paths these projectiles take and share your findings with the class.

Materials Needed

1. A notebook and pen/pencil (for recording your lists)

2. A whiteboard or large paper (optional, for group feedback) What to do

1. In your pairs, brainstorm and list as many sports as you can think of. Aim for at least 5-10 different sports.

2. For each sport on your list, identify the objects that act as projectiles.

Write down the projectile associated with each sport next to the sport’s name in your notebook.

3. Discuss with your partner how each projectile moves through the air.

Consider:

a. The shape of the path (e.g., straight line, curve, parabolic).

b. Factors that might affect the path (e.g., angle of launch, speed, air resistance).

c. Make notes on these observations next to each projectile in your list.

4. Once you have completed your list and descriptions, prepare to share your findings with the class.

5. One pair at a time should present their list of sports and projectiles to the class for feedback Engage in a whole class discussion and discuss how different factors might influence projectile motion in various sports.

6. Watch the video linked below demonstrating some examples of projectile motion in sports.

Projectile Motion Examples Sports

Note: you can find some examples of the applications of projectiles in Annex B.

Activity 5.3 Investigating Projectile Motion with Launchers

Objective: Explore how the angle of launch affects projectile motion by measuring key quantities such as time of flight and range.

Materials Needed

1. A small catapult or slingshot

2. Projectiles (e.g., small balls or bean bags)

3. Rulers (to measure distance)

4. Protractors (to measure launch angles)

5. Stopwatches (to measure time)

6. A notebook and pen/pencil (for recording data) What to do

1. Place your catapult or slingshot on a flat surface. Ensure it is stable and secure.

2. Use the protractor to set the launcher at different angles. Suggested angles include 15°, 30°, 45°, 60°, and 75°. Record each angle in your notebook.

3. For each angle:

a. Pull back the launcher to a consistent position for uniform launch conditions.

b. Release the projectile while simultaneously starting the stopwatch.

4. Stop the stopwatch as soon as the projectile lands on the ground. Record this total time of flight in your notebook.

5. Measure the horizontal distance from the base of the launcher to where the projectile lands using a ruler. Record this distance as the range for each launch angle.

6. Repeat steps 3 to 5 for each launch angle you set in step 2.

7. Firstly, use the following formula to calculate initial velocity (u) based on your measured time of flight (T):

u = gT____ 2sinθ Where:

g is 9.81 ms⁻²(acceleration due to gravity), θ is the launch angle in degrees.

8. Secondly, use this formula to calculate initial velocity (u) based on your measured range (R):

u = √Rg_ sin2θ

9. Write down both calculated values of initial velocity for each angle in your notebook.

10. Compare the initial velocities obtained from both methods (time of flight and range). Discuss any differences you observe and consider possible reasons for these discrepancies.

11. Reflect on how changing the launch angle affected your measurements of the time of flight, range, and calculated initial velocity. Consider what you learned about projectile motion through this experiment.

Activity 5.4 Projectile Motion Simulation Experiment

Objective: To explore the principles of projectile motion by using simulations to change variables such as initial velocity, launch angle, and height of the launcher. You will measure the resulting time of flight and range, compare simulation results with theoretical calculations, and discuss any discrepancies.

Materials Needed

1. A computer or tablet with internet access

2. Access to a projectile motion simulation (e.g., PhET Interactive Simulations: https://phet.colorado.edu/en/simulations/projectile-motion)

3. Calculator (for theoretical calculations)

4. Notebook or digital document for recording data and observations

5. Graph paper (optional for plotting results) What to do

1. Open a web browser and navigate to the PhET Interactive Simulations website. Click on the Projectile Motion simulation.

2. Spend a few minutes exploring the interface of the simulation. Understand how to adjust parameters such as:

a. Initial Velocity: The speed at which the projectile is launched.

b. Launch Angle: The angle at which the projectile is launched relative to the horizontal.

c. Height of Launcher: The vertical height from which the projectile is launched (you can set this to zero).

3. Create a table in your notebook to predict your results using the formulae in step 4:

Initial Velocity

(m/s) Launch Angle (°)

Theoretical Time of

Flight (s) Theoretical Range

(m)

4. For each set of parameters, decided by you and filled in to the first two columns of your table, calculate the theoretical time of flight and range using the following formulas:

a. Time of Flight (T): T = 2usinθ______ g

b. Range (R): R = u²sin2^(θ)_ g Where:

u = initial velocity θ = launch angle in radians g = acceleration due to gravity (approximately 9.81 m/s²)

5. Set Parameters on the Simulation

a. Choose the initial velocity for your projectile (e.g., 10 m/s, 20 m/s, etc.).

b. Select a launch angle (e.g., 30°, 45°, 60°).

c. Set the height of the launcher to zero.

6. Launch the projectile by clicking on the appropriate button in the simulation.

7. Observe and record the following outcomes:

a. ime of Flight: How long the projectile stays in the air.

b. Range: The horizontal distance travelled by the projectile.

8. After calculating theoretical values, compare them with your simulation results. Note any discrepancies between your calculated values and those obtained from the simulation.

9. Reflect on possible reasons for any differences between theoretical calculations and simulation outcomes:

a. Consider factors such as air resistance, inaccuracies in angle measurement, or limitations of the simulation.

b. Discuss how real-world conditions might affect projectile motion compared to ideal calculations.

Activity 5.5 Sports Projectiles Poster Project

Objective: To explore the concept of projectiles in various sports by creating informative posters or collages that showcase different sports and the projectiles used in each.

Materials Needed

1. Large poster boards or sheets of paper

2. Markers, coloured pencils, or crayons

3. Magazines or printed images related to sports

4. Glue or tape

5. Ruler

6. Access to research materials (books, internet, etc.)

7. Scissors What to do

1. Organise yourselves into small groups of no more than five.

2. As a group, select 2-3 different sports to focus on for your poster. Consider including a variety of sports that use different types of projectiles, such as:

a. Baseball (baseball)

b. Basketball (basketball)

c. Football (football)

d. Tennis (tennis ball)

e. Golf (golf ball)

f. Archery (arrow)

3. Research the projectiles used in each chosen sport. Gather information about:

a. The type of projectile used.

b. How the projectile is used in the sport.

c. Any interesting facts about the projectile’s design or performance.

4. Discuss how you want to organise your poster. Consider dividing the poster into sections for each sport, including:

a. The name of the sport.

b. An image of the projectile.

c. Key facts about the projectile.

5. Sketch a rough layout on a separate piece of paper before starting on the poster board.

6. Using your large poster board or paper, start creating your poster:

a. Write the names of each sport in bold letters.

b. Draw or paste images of the projectiles next to their respective sports.

c. Include key facts and information about each projectile in clear, legible text.

d. Make sure your poster is colourful and visually appealing.

7. Once all groups have completed their posters, display them around the classroom.

8. As a class, participate in a gallery walk where you can move around the room to view each group’s poster. Take notes on interesting facts you learn from other groups’ posters.

Activity 5.6 Calculating Quantities associated with projectiles Study the worked examples below carefully before attempting the example questions that follow.

Worked Example 1

Solved problems

1. A ball is launched from the ground with an initial velocity of 30 m/s at an angle of 40⁰to the horizontal. Assume g=9.8 m/s². Find the:

a. maximum height reached by the ball.

b. time taken to reach the maximum height.

c. total time of flight of the ball.

d. range or horizontal distance travelled by the ball.

Solution

Given data: u = 30 ms⁻¹, θ = 40° , g = 9.8 m s⁻²At max height: v = 0 m s⁻¹a. H = u²sin²θ_ 2g H = 30²sin²40_ 2(9.8) H = 900 × 0.4132/19.6 = 18.973 m

b. t = usinθ_____ g t = 30 sin40/9.8 t = 30 × .6428/9.8 = 1.968 s

c. t_(f) = 2t = 2(1.968) = 3.935 s R = utcosθ = 30cos40 × 3.935 = 90.4 m

2. A cannon fires a projectile at an initial velocity of 50 m/s at an angle of 60⁰ to the horizontal from a cliff 20 m above the ground. Assume g=9.8 m/s².

Find:

a. maximum height of the projectile above the ground.

b. total time of flight until the projectile hits the ground.

c. range (horizontal distance) from the base of the cliff where the projectile lands.

Solution

Given data: u = 50 ms⁻¹, θ = 60° , g = 9.8 m s⁻²At max height: v = 0 m s⁻¹a. Step 1: find the maximum height from the point of projection hₘₐₓ.

hₘₐₓ = u²sin²θ_ 2g = 50²sin²60_ 2(9.8) = 95.66 m

Step 2: add the initial height from the ground (20 m) to hₘₐₓ and call it Hₘₐₓ Hₘₐₓ = hₘₐₓ + 20 Hₘₐₓ = 95.66 + 20 = 115.66 m So the maximum height of the projectile from the ground is 115.66 m.

b. The time of flight here is the time the projectile took to the maximum height and fell to the ground. So, you need to find the time it took to travel the upward and downward distances separately and add them.

Step 1: find the time taken to travel hₘₐₓ = 95.66 m, you may call it tᵤₚ tᵤₚ = u _ _sinθ _____ g = 5_0 sin60 _______ 9.8 = 4.415 s

Step 2: find the time taken to land on the ground from the maximum height ( Hₘₐₓ,), you may call it t_(down).

Hₘₐₓ = u t_(down) + 1/2 g t2/(down), As the projectile begins to come down, its initial velocity u=0, so Hₘₐₓ = (0) t_(down) + 1/2 g t2/(down), Hₘₐₓ = 1/2 g t2/(down), t_(down) = √____ 2 Hₜₒₜₐₗ_____ g = √_______ 2(115.66)_______ 9.8 = 4.485 s

Step 3: add t_(down) to tᵤₚ to get the time of flight t_(f) t_(f) = t_(down) + tᵤₚ t_(f) = 4.418 + 4.845 = 9.263 s

c. R = (ucos θ) × t_(f) R = (50cos 60)(9.263) R = 231.575 m Practice questions

1. A football is kicked horizontally from a height of 10 m with an initial velocity of 5 m/s.

a. How long does it take to hit the ground?

b. How far does it travel horizontally before hitting the ground?

(Assume g = 9.8 m/s²)

2. A ball is launched with an initial velocity of 25 m/s at an angle of 45⁰.

Calculate the range of the projectile.

(Use g=9.8 m/s²)

3. A ball is kicked with an initial velocity of 15 m/s at an angle of 45⁰.

a. How long does the ball stay in the air?

b. How far does it travel horizontally before hitting the ground?

(Assume g=10 m/s²) FRICTION When you push a desk on a rough concrete floor, you notice significant resistance to your effort. In contrast, pushing the same desk on a tiled floor requires less effort.

Now, if you give the desk a push and let it move freely on the concrete floor, and then do the same on the tiled floor with a similar push, you will observe that the desk moves more freely and farther on the tiled floor. This happens because the smoother tiled surface offers less resistance compared to the rough concrete floor.

In both scenarios, there is a force resisting the initial push on the desk. After the push, the desk’s motion slows down or decelerates more quickly on the rough surface than on the smooth surface. This resisting force, which opposes motion or decelerates a moving object, is called friction. Friction arises between surfaces in contact and depends on the nature of those surfaces and their relative motion.

Friction is the force that opposes the relative motion or the tendency of such motion of two surfaces in contact.

Figure 5.3: Friction between two surfaces in contact Types of friction

1. Static friction: This is the force that resists the initial sliding motion between two surfaces in contact. It is also referred to as limiting friction because it represents the maximum force that must be overcome to start motion. In the scenarios above, static friction is the reason for the difficulty in getting the desk to start moving across the concrete floor.

2. Dynamic friction: This is the force between surfaces in contact that tends to decelerate or resist the relative motion of the bodies. In the scenarios above, this force is responsible for slowing down the moving desk and eventually bringing it to a stop.

In the process of moving a stationary object, the applied force must exceed the limiting friction before the body can accelerate or move.

The magnitude of the frictional force depends on:

1. The coefficient of friction, which in turn depends on the nature of the surfaces.

2. The normal reaction R, which is the force that acts perpendicular to the surfaces, pressing them together.

Note that, neither the area of the surfaces nor their relative speed of motion affects the magnitude of the frictional force.

The coefficient of friction (μ) This is a number that quantifies the ratio of the frictional force F resisting motion to the normal reaction force R pressing the two surfaces together. It varies depending on the materials in contact and their surface conditions.

Mathematically, μ= F/R This is a dimensionless constant and bears no SI unit. It is unique for any given pair of surfaces.

Types of Coefficients of Friction

1. Coefficient of static friction (μₛ), this:

a. Represents the friction when an object is at rest.

b. Determines the force required to initiate motion.

c. Typically, greater than the kinetic coefficient.

2. Coefficient of kinetic friction( μₖ), this:

a. Represents the friction when an object is already in motion.

b. Determines the force required to keep maintain or sustain the object’s motion.

Activity 5.7 Exploring the Role of Friction in Everyday Life Objective: identify and discuss scenarios in everyday life where friction is advantageous, as well as situations where friction causes problems.

Materials Needed

1. Notebook and pen/pencil (for recording ideas)

2. Whiteboard or large paper (optional, for group brainstorming)

3. Markers (optional, for writing on the whiteboard) What to do

1. Take a few minutes to think about your daily life. Write down at least three scenarios where friction is beneficial. Consider areas such as:

a. Transportation (cars, bicycles, shoes)

b. Sports (gripping surfaces, traction)

c. Everyday tasks (writing, cooking, cleaning)

2. Write down at least three scenarios where friction causes problems or challenges. Think about situations like:

a. Wear and tear on materials (tyres, machinery)

b. Difficulty in movement (slipping on ice, heavy objects)

c. Excessive heat generation (brakes in vehicles)

3. Find a partner and share your lists. Discuss your examples of advantageous friction and problematic friction. As you discuss, consider questions like:

a. Why is friction important in the scenarios you identified?

b. What specific problems does friction create in the negative examples?

c. Are there any solutions or methods to reduce the negative effects of friction?

4. After your pair discussions, come together as a class, and share your examples of advantageous friction with the class.

5. Share your examples of problematic friction. Discuss these as a class and brainstorm possible solutions for each issue raised.

6. As a class, create two summary lists on the whiteboard or large paper:

a. List all the scenarios discussed where friction is beneficial.

b. List all the scenarios discussed where friction causes issues along with potential solutions.

7. Watch the video linked below demonstrating some examples of friction in everyday life.

https://youtu.be/qy-EJRDyt-A?si=lK8aOthIk-tGtgyY

8. See Annex B for some examples of friction, including its disadvantages and steps which can be taken to reduce it.

Activity 5.8 Measuring Static and Dynamic Friction

Objective: Conduct experiments to measure the forces of static and dynamic friction using a spring scale on various surfaces.

Materials Needed

1. Spring scale (capable of measuring in Newtons)

2. Various surfaces (e.g., rubber mat, wooden board, metal sheet)

3. Object to test (e.g., a block or cart)

4. Ruler (for measuring dimensions if needed)

5. Notebook and pen/pencil (for recording data)

6. Weights (optional, for varying the mass of the object) What to do

1. Choose one of the surfaces (e.g., rubber mat) and place it flat on a table or the floor. Place the object you are testing on top of this surface.

2. Attach the spring scale to the object. Gradually pull the object using the spring scale until it just starts to move.

Figure 5.4: Spring balance attached to an object

3. Record the maximum force indicated on the spring scale just before the object begins to slide. This value is the static friction force for that surface.

Write down your measurements in your notebook.

4. Once the object is in motion, continue pulling it at a constant speed.

5. Record the force indicated on the spring scale while maintaining that constant speed. This value is the dynamic (kinetic) friction force for that surface. Write this measurement down in your notebook as well.

6. Change to a different surface (e.g., wooden board) and repeat steps 2,3, 4 and 5.

7. Record your measurements for static and dynamic friction forces in your notebook for each surface.

8. Continue this process for all surfaces you have available (e.g., rubber, wood, metal).

9. Organise your recorded data into a table with the following format:

Surface Type Static Friction Force (N) Dynamic Friction Force (N) Rubber Wood Metal

10. Compare the static and dynamic friction forces across different surfaces.

11. Look for trends such as which surface has the highest or lowest friction forces. Consider questions like:

a. How does surface texture affect friction?

b. Is there a significant difference between static and dynamic friction on each surface?

Activity 5.9 Investigating Static Friction with Interactive Simulations

Objective: Use interactive simulations to explore static friction by adjusting the weight of an object.

Materials Needed

1. A computer or tablet with internet access

2. Access to the PhET Forces and Motion simulation or a similar interactive simulation that allows exploration of static friction (https://phet.colorado.

edu/sims/html/forces-and-motion-basics/latest/forces-and-motion- basics_all.html)

3. A notebook and pen/pencil (for recording data)

4. Graph paper or graphing software (optional, for plotting your graph) What to do

1. Open your web browser and go to the PhET Forces and Motion simulation or another interactive simulation focused on friction.

2. Spend a few minutes exploring the interface. Identify where you can adjust the weight of the object and observe how it interacts with the surface.

(turn on the values button)

3. Choose an object (like a wooden box) to work with in the simulation.

4. Record the mass of the object in your notebook.

5. Gradually apply a force to the object until it just begins to move. This maximum force is the static friction force (or the displayed frictional force value)

6. Record this force in your notebook along with the corresponding mass and normal force (which can be calculated based on weight).

7. Create a table in your notebook with the following format:

Trial Mass (kg) Normal Force (N) Static Friction Force (N) 1 2 3 4 5

8. The normal force can be calculated using the formula:

Normal Force = Mass × g Where g is approximately 9.81 ms⁻². Ensure you calculate this for each mass you record.

9. Change the mass using other objects or combination of objects (90, 100 kg, 140 kg, 150 kg, 180 kg) and repeat steps 5 to 7. Record all measurements in your table for comparison.

10. Using your completed table, plot a graph of Static Friction Force against Normal Force.

11. Examine your graph for trends and relationships between normal force and static friction force. Consider questions like:

a. How does increasing weight affect static friction?

b. Is there a consistent relationship between normal force and static friction?

Activity 5.10 Calculating Coefficient of friction Study the worked examples below carefully before attempting the example questions that follow.

Worked Example 1

A 10 kg block rests on a horizontal surface. The coefficient of static friction is 0.4. What is the minimum horizontal force required to start moving the block?

[g=9.8 m/s]

Solution

Step 1: Identify the given data:

mass = 10 kg, μₛ = 0.4 , g=9.8 m/s

Step 2: Introduce the formula for static friction The block is horizontal and the only normal force is its weight.

So R = mgμₛ = Fₛ_ R = Fₛ_ mg

Step 3: Substitute the given data and calculate the static friction 0.4 = Fₛ_ 10 × 9.8 Fₛ = 0.4 × 98 = 39.2 N

Step 4. The minimum force applied must be equal to the static friction.

Therefore, f = Fₛ = 39.2 N

Worked Example 2

You pushed your little brother horizontally on a wooden box with a force of 70 N. If the mass of the box is 2 kg and that of your brother is 15 kg, calculate the net force that kept your brother in motion on the box. The coefficient of kinetic friction is 0.2 and g = 10 ms⁻². Would your brother remain at a constant speed or accelerate?

Solution

Step 1: Identify the given data Mass of box m_(bx) = 2 kg, the mass of brother m_(bro) = 15kg , g = 10 ms⁻². μₖ = 0.2

Step 2: Calculate the total mass m m = m_(bx) + m_(br) = 2 + 15 = 17 kg

Step 3: Calculate the normal reaction which is the total weight of the moving system of masses.

R = mg = 17 × 10 = 170 N

Step 4: Calculate the kinetic friction Fₖ Fₖ = μₖ × R = 0.2 × 170 = 34 N

Step 5: subtract the frictional force Fₖfrom the applied force F to determine the net force Fₙₑₜ = F − Fₖ = 70 − 34 = 36 N Therefore, the force that kept the system moving is 36 N. The brother would accelerate as the net force is not zero.

Worked Example 3

A 5 kg block is placed on a horizontal surface. The coefficient of static friction between the block and the surface is 0.4, and the coefficient of kinetic friction is 0.3. A horizontal force of 18 N is applied to the block. Will the block move?

[g=9.6]

Solution:

Step 1: Identify the given data:

Mass of block m=5 kg, μₛ = 0.4 , μₖ = 0.3 , Let’s call the applied force F.

F = 18 N

Step 2: Find the static friction Fₛ For a horizontal body like this particular one, the only normal reaction force is the weight of the block. So R = mg = 5 × 9.8 = 49 N μₛ = Fₛ_ R 0.4 = Fₛ_ 49 Fₛ = 0.4 × 49 = 19.6 N

Step 3: Compare the static friction to the applied force to determine if the applied force is sufficient to overcome the friction and start the motion.

F = 18 N, Fₛ = 19.6 N Conclusion: since F < Fₛ, the block will not move.

Practice Problems

Now, using the worked example as a guide, solve the following problems individually or in groups.

1. A block of mass 20 kg rests on a horizontal surface. It takes a horizontal force of 50 N to just start moving the block. Find the coefficient of static friction. [g=10 ms²]

2. A 20 kg box is pulled on a horizontal surface with a constant speed using a force of 80 N. Find the coefficient of kinetic friction. [g=10 ms²]

3. A 15 kg crate is pushed across a rough surface with an acceleration of 2 m/s².

The coefficient of kinetic friction between the crate and the surface is 0.25.

Find the applied force. [g=10 ms²]

Circular motion is a type of motion in which an object moves around a fixed point while maintaining a constant distance from that point. The object does not need to complete an entire circle to be considered in circular motion. As the body moves in a circle, its change in position is as a result of its radius vector sweeping through an angle. The angle through which the radius vector sweeps in order for the to change its position in along the arc is known as the angular displacement.

Figure 5.5: Circular motion Examples of circular motion

1. A Ball on a String: Swinging a ball on a string in a circle creates circular motion, where the tension in the string provides the centripetal force to keep the ball moving in its path.

2. Electrons in Atoms: In simplified atomic models, electrons are often depicted as moving in circular orbits around the nucleus, due to the attraction between the negatively charged electrons and the positively charged nucleus.

3. Planets Orbiting the Sun: The planets move in nearly circular orbits around the Sun, held in place by the Sun’s gravitational pull acting as the centripetal force.

4. A Car negotiating a Curve: When a car makes a turn, it follows a curved path, with the friction between the tires and the road providing the centripetal force to keep it on track.

Activity 5.11 Watch a video demonstrating some every day examples of circular motion Objective: Watch a video demonstrating everyday examples of circular motion and analyse the concepts presented.

Materials Needed

1. Access to a computer or tablet with internet access

2. Projector or large screen (if watching as a group)

3. Paper and pens for notes

4. Access to the video link: 5 Non Uniform Circular Motion Examples In Physics & Daily Life What to do

1. Form small groups of no more than five.

2. Watch the Video

a. Open the provided video link on your device or project it on a larger screen for the group.

b. Watch the video titled “5 Non-Uniform Circular Motion Examples In Physics & Daily Life,” which showcases various examples of circular motion in everyday life.

3. While watching, take notes on the following:

a. The examples of circular motion presented in the video (e.g., spinning top, frisbee, ice skater, car on a curved road, roller coaster loop).

b. Key points about how each example demonstrates circular motion.

c. Any specific terms or concepts related to circular motion that are mentioned in the video.

4. After watching the video, come together in your groups to discuss your notes. Use the following prompts to guide your discussion:

a. What were your favourite examples of circular motion from the video? Why?

b. How do you think understanding circular motion is important in real life?

c. Can you think of other examples of circular motion that were not mentioned in the video?

5. Each group should prepare to share one key insight or interesting fact they learned from the video with the class.

6. Discuss how these concepts apply to real-world scenarios and why they are significant in understanding physics.

Angular displacement Angular displacement tells us how much an object has turned or rotated about a point. Instead of measuring this distance in meters (like we would for straight lines), we measure it in angles.

Figure 5.6: Schematic diagram of circular motion In the diagram (Figure 5.6), if an individual walks from one point to another along the circumference of a circle, they will have undergone an angular displacement.

The distance travelled depends on the radius of the circle and can be expressed as:

θ = s_ r s = θr Where, θ is the angular displacement, s is the distance travelled by the body, and r is the radius of the circle along which it is moving.

Note that:

2π = 360° = 1 revolution Angular velocity Angular velocity is the rate of change of angular displacement with time.

ω = θ_ t where θ = angular displacement, t = time, ω = angular displacement but θ = s_ r ω = s__ tr From linear motion v = s_ t ⇒ ω = v_ r v = ωr The relationship between angular velocity ω and linear velocity v is v = ωr Angular velocity ω is also given by ω = 2π__ T = 2πf where f is the frequency and T is the period.

Frequency is the number of complete revolutions made in a second. The period is the time taken for one complete revolution.

Angular Acceleration

Angular acceleration (α) is the rate of change of angular velocity with time.

α = ω__ t but ω= v_ r α = v__ rt Also, from linear motion α = v_ t ∴ α = α__ r a = αr The relationship between angular acceleration and linear acceleration is a = αr Centripetal Force Bodies moving along a curved path or in a circle are maintained in their trajectory by a force directed toward the centre of the circle. This force, known as centripetal force, prevents the body from moving in a straight line due to its inertia. If the centripetal force is removed, the body will no longer follow the curved path and will move in a straight line tangential to the circle at the point where the force was removed. For example, if you whirl a stone around your head on a string and then release it, you have removed the centripetal force and the stone will move off in a straight line from the point of release.

Centripetal force is a force that is exerted on a body moving in a circle for the body to be kept in its path.

Centripetal force is given by F_(c)= m v²____ r

Activity 5.12 Exploring Circular Motion with a Fan/turntable

Objective: Use a fan to demonstrate circular motion.

Materials Needed

1. A standard electric fan (preferably with adjustable speed settings)/ turntable

2. Stopwatch

3. Ruler

4. Notebook and pen/pencil (for recording data) What to do

1. Place the fan on a stable surface and ensure it is plugged in and functioning properly. Make sure the fan is set to a low or medium speed to start.

2. Identify a point on the fan blades where you will measure the radius. Use the ruler to measure the distance from the centre of the fan (the axis of rotation) to the tip of one of the blades. This distance is your radius (r).

Record this value in your notebook.

3. Turn on the fan at a low or medium speed.

4. Use a stopwatch to time how long it takes for the fan to complete several revolutions (e.g., 5 or 10). Start timing as you complete your first full revolution and stop when you reach your last revolution. Record this total time in your notebook.

5. Calculate the period (T), which is the time taken for one complete revolution:

T = Total time________________ number of revolutions

6. Calculate frequency, which is the number of revolutions per second:

f = 1/T

7. Calculate angular velocity (ω) using the formula:

ω = 2πf

Note that π is approximately 3.14.

8. Measure angular displacement for multiple revolutions, count how many times the object completes a full rotation during your timing and multiply by 360 degrees:

Angular Displacement = Number of Revolutions × 360°

Activity 5.13 Investigating Centripetal Force through Circular Motion Objective: To swing a small object tied to a string in a circular path to illustrate centripetal force and to measure the radius of the circular path and the time taken for several revolutions to calculate the period, frequency, angular velocity, and centripetal force.

Materials Needed

1. A small object (e.g., a rubber ball or a small weight)

2. A length of string (about 1-2 meters)

3. Stopwatch (or a timer on your phone)

4. Ruler or measuring tape (to measure the radius)

5. Notebook and pen/pencil (for recording data) What to do

1. Tie one end of the string securely to the small object.

2. Measure the length of the string from the centre of the object to the point where you will hold it. This length will be your radius (r) of the circular path.

3. Use a ruler or measuring tape to measure the length of the string (the radius). Record this value in your notebook.

4. Hold the other end of the string firmly in your hand and swing the object in a horizontal circular path. Make sure to keep the string taut while swinging. Try to maintain a consistent speed as you swing the object.

5. Use a stopwatch to time how long it takes for the object to complete several revolutions (e.g., 5 or 10). Start timing as you complete your first full revolution and stop when you reach your last revolution. Record this total time in your notebook.

6. Calculate the period (T), which is the time taken for one complete revolution.

7. Calculate frequency, which is the number of revolutions per second.

8. Calculate angular velocity (ω) .

9. Calculate centripetal force using the formula F_(c) = m ω²r Where:

m is the mass of the object (you can weigh it if needed), ω is angular velocity, r is the radius you measured.

10. Extension: Investigate how changing variables like mass or radius might affect centripetal force. Consider questions such as:

a. What happens to centripetal force if you increase speed?

b. How does increasing radius affect angular velocity?

Activity 5.14 Investigating Circular Motion using an Online Simulation Objective: Use an online simulation to explore the variables related to circular motion, including radius, speed, and centripetal force.

Materials Needed

1. Access to a computer or tablet with internet access

2. Paper and pens for notes

3. Access to the simulation: Uniform Circular Motion Simulation What to do

1. Form small groups of no more than five. Each group will work together to explore the simulation.

2. Click on the provided link to access the Uniform Circular Motion simulation.

3. Once the simulation loads, familiarise yourself with the interface and available controls.

4. Begin by observing the default settings of the simulation. Take note of the initial values for radius, speed, and centripetal force displayed.

5. Investigate Variables

a. Change Radius

i. Use the controls to adjust the radius of the circular path.

ii. Observe how changing the radius affects the speed and centripetal force required to maintain circular motion.

iii. Take notes on your observations.

b. Change Speed

i. Adjust the speed of the object moving in a circle.

ii. Record how increasing or decreasing speed impacts centripetal force and any other observed variables.

c. Analyse Centripetal Force

i. With different combinations of radius and speed, note how centripetal force changes.

ii. Discuss within your group why these changes occur based on your understanding of circular motion.

6. Create a table in your notes to summarise your findings:

a. Record different combinations of radius, speed, and centripetal force.

b. Note any patterns or relationships you observe between these variables.

7. After exploring various settings in the simulation, come together as a group to discuss your findings. Use these prompts to guide your discussion:

a. What patterns did you notice between radius, speed, and centripetal force?

b. How does increasing speed affect the amount of centripetal force needed?

c. What practical applications can you think of that relate to circular motion (e.g., cars on a curved road, satellites orbiting a planet)?

Activity 5.15 Calculating Angular velocity, Angular displacement Study the worked examples carefully below before attempting the example questions that follow

Worked Example 1

A bicycle wheel with a radius of 0.5 meters completes 15 revolutions in 30 seconds. Calculate:

i. the angular velocity of the wheel.

ii. the linear velocity of a point on the rim of the wheel.

iii. The frequency of revolution

Solution

Step 1. Identify given variables θ = 15 rev, r = 0.5 m, t = 30 s, ω = ?

v = ?

f = ?

Step 2. Determine angular velocity from revolution made in the given time 30 s = 15 revs, 1 s = 1 s/30 s × 15 revs = 0.5 revs, ω = 0.5 rev/s 1 rev = 2π rad, 0.5 rev = 0.5 rev/1 rev × 2π rad = π rad, ω = 0.5 rev/s = π rad/s

Step 3. Determine linear velocity v = ω r = π rad/s x 0.5 m, π = 3.142, v = 3.142/s × 0.5 m = 1.571 m/s

Step 4. Determine frequency revolution ω = 2πf π rad/s = 2πf f = 0.5 Hz

Worked Example 2

A rotating fan blade starts from rest and reaches an angular velocity of 20 rad/s in 5 seconds.

Calculate:

i. the angular acceleration of the fan blade.

ii. the total angular displacement of the blade during this time.

Solution

Step 1 Identify the given variables ωᵢ= 0, ω_(f)= 20 rad/s, t = 5 s, α = ? θ = ?

Step 2: Introduce the formula for angular acceleration α = ∆ ω___ t = ω_(f)− ωᵢ_____ t = 20 − 0/5 = 4 rad/s²Step 3: Use an equation of motion to get your answer θ = ωᵢt + α t²___ 2 = 0(5) + 4( 5²)____ 2 = 50 rad Practice Problems Now, using the worked example as a guide, solve the following problems individually or in groups.

1. A wheel rotates at a constant angular velocity of 6 rad/s. How much angular displacement does the wheel undergo in 10 seconds?

2. A merry-go-round rotates with an angular velocity of 2 rad/s. How many complete revolutions does it make?

3. A rotating platform has a radius of 2 m and completes 15 revolutions per minute.

a. Calculate the angular velocity in radians per second.

b. If a 2 kg object is placed on the edge of the platform, find the centripetal force acting on it.

Have you ever wondered why motorbike riders tilt their bikes when negotiating a curve? Have you also noticed that, despite tilting, they sometimes still skid while navigating a curve?

Roads are banked to reduce the risk of skidding and to ensure safe navigation of curves. Banking involves raising the outer edge of the road at a specific angle, which helps generate a component of the normal force that acts as the necessary centripetal force.

This design enables vehicles to negotiate curves more safely by reducing reliance on friction.

Activity 5.16 Watch a video demonstrating the effect of banking Objective: Watch a video that demonstrates the concept of banking on roads.

Materials Needed

1. Access to a computer or tablet with internet access

2. Projector or large screen (if watching as a group)

3. Paper and pens for notes

4. Access to the video link: Banking of Roads What to do

1. Form small groups of no more than five.

2. Watch the Video

a. Open the provided video link on your device or project it on a larger screen for the group.

b. Watch the video titled “Banking of Roads,” which explains how banking helps vehicles safely navigate turns by examining the forces at play.

3. While watching, take notes on the following:

a. The relationship between centripetal force and friction when a vehicle takes a turn.

b. How banking affects the maximum allowed speed while turning.

c. Key equations presented in the video, such as those relating to banking angle, radius, and velocity.

4. After watching the video, come together in your groups to discuss your notes. Use the following prompts to guide your discussion:

a. What are the main forces acting on a vehicle during a banked turn?

b. How does banking increase safety compared to flat turns?

c. What factors can affect the effectiveness of banking (e.g., weather conditions)?

d. Can you think of real-life examples where banking is applied (e.g., highways, racetracks)?

Figure 5.7: Image of a banked road

Figure 5.8: Schematic diagram of a vehicle negotiating a curve In the diagram above, the weight (mg) of the car which acts vertically downwards is balanced by N cosθ This gives us the expression N cos θ = mg ----------------- eqn (1) The centripetal force F_(C)= m v ²____ r which is directed towards the centre of the circle, is what keeps the car along its circular path. It is provided by N sin θ because N sin θ is directed towards the centre.

This gives us the expression N sin θ = m v²____ r ---------------- - eqn (2) Divide eqn (2) by eqn (1) N sinθ_____ N cosθ = m v²____ r____ mg tanθ = v²__ rg where θ = banking angle, v = maximum speed that you can negotiate the curve without skidding, r = radius of the curve/track, g = acceleration due to gravity.

Notably, this maximum speed is higher than it would have been without the banking.

Activity 5.17 Calculating Banking Angle

Study the worked example carefully below before attempting the example questions that follow.

Worked Example 1

A curve on a road is designed to allow cars to safely pass through at a speed of 20 m/s. If the radius of the curve is 200 m, what is the angle at which the road should be banked to prevent the car from sliding?

Step-by-Step Solution

Step 1: Identify Given data v=200 m/s, r = 200 m, g=10 m/s²Step 2: Introduce the formula for banking angle tanθ = v²_ rg

Step 3: substitute the given values into the formula and calculate tanθ = 20²_ 200 × 10 = 400/2000 = 0.2

Step 4: Make the angle the subject and calculate θ = tan⁻¹(0.2) = 11.31° Practice Problems Now, using the worked example as a guide, solve the following problem individually or in groups.

1. A car moves on a curved highway at a speed of 25 m/s on a curve of radius 50 m. If the acceleration due to gravity is 9.8 m/s², calculate the banking angle required for the curve.

2. A curve on a road is banked at an angle of 15°. If the radius of the curve is 100 m, what is the maximum speed a car can travel without relying on friction? (g= 10 m/s²)

3. A racing bike moves around a banked track with a speed of 30 m/s. If the track is designed with a banking angle of 40°, determine the radius of curvature required to keep the rider safe. (g= 9.8 m/s²)

4. A car takes a turn on a road banked at 25° with a curve radius of 80 m.

Determine the maximum speed at which the car can move without relying on friction. (g= 9.8 m/s²) The centrifuge Mixtures can be separated by spinning them at very high speeds using a device called a centrifuge. During this process, a strong outward force acts on the components of the mixture, pushing substances with higher densities toward the outer edge, while substances with lower densities remain closer to the centre. This outward force is often referred to as centrifugal force.

Applications of a Centrifuge

Centrifuges have a wide range of applications across various fields due to their ability to separate mixtures based on density. Here are some key applications:

1. Biological and medical applications

a. Blood component separation:

i. Blood Banks: Separating red blood cells, plasma, and platelets from donated blood.

ii. Clinical Diagnostics: Preparing blood samples for tests by separating serum or plasma from whole blood.

b. Cell fractionation: Isolating different components of cells, such as nuclei, mitochondria, and lysosomes, for research purposes.

c. DNA/RNA purification: Isolating nucleic acids from cells or tissues for genetic analysis.

d. Protein separation and purification: Separating proteins based on size and density for biochemical studies.

f. Urine analysis: Concentrating cells and other components from urine samples for diagnostic tests.

2. Industrial Applications

a. Food and beverage industry

i. Cream Separation: Separating cream from milk in dairy processing.

ii. Juice Clarification: Removing pulp and other solids from fruit juices.

iii. Sugar Production: Separating sugar crystals from molasses in sugar refineries.

b. Oil industry

i. Waste Oil Recycling: Removing contaminants from used oil for reuse.

ii. Oil Extraction: Separating oil from other substances, such as in the production of essential oils.

c. Water Treatment: Clarifying water by removing suspended solids and other impurities.

3. Chemical and Pharmaceutical Applications

a. Precipitate Separation: Isolating solid precipitates from liquid mixtures in chemical reactions.

b. Purification of Pharmaceuticals: Separating and purifying drugs and other chemical compounds.

c. Nanoparticle Separation: Isolating nanoparticles based on size and density for research and industrial applications.

4. Environmental Applications

a. Soil Analysis: Separating soil particles for analysis of composition and contaminants.

b. Wastewater Treatment: Removing sludge and other solid particles from wastewater before discharge or reuse.

5. Research and Development

a. Laboratory Research: Performing various types of separations in biochemical, molecular biology, and clinical research.

b. Material Science: Studying the properties of materials by separating their components.

6. Other Applications

a. Aerospace: Training astronauts and pilots by simulating high-gravity conditions in human centrifuges.

b. Forensic Science: Analysing samples from crime scenes, such as separating components of biological samples.

c. Cosmetic Industry: Formulating and testing cosmetic products by separating ingredients.

Activity 5.18 Investigating centrifuges Through Video and Hands-On Construction Objective: Watch videos to understand the principles of centrifugation and then construct a simple centrifuge using a salad spinner to observe the effects of centrifugal force.

Materials Needed

1. Access to a computer or tablet with internet access

2. Projector or large screen (if watching as a group)

3. Video Link:

a. Centrifugation Process

b. Proper Use of a Centrifuge

4. Paper and pens for notes

5. A salad spinner

6. Small containers (like test tubes or small cups)

7. Water or a mixture (e.g., muddy water or coloured water)

8. Optional: food colouring, soil, or small particles for separation What to do Part A: Watch the Videos

1. Form small groups of no more than five.

2. Open the first video link on your device or project it on a larger screen for the group.

3. Take notes on:

a. How centrifugation works to separate components based on density.

b. The materials used in the demonstration and the results observed after centrifugation.

c. Key concepts related to centrifugal force and its dependence on mass and speed of rotation.

4. Open the second video link on your device or project it on a larger screen.

5. Take notes on:

a. The proper techniques for using a centrifuge.

b. The importance of balancing test tubes in the centrifuge.

c. Any safety precautions mentioned in the video.

Part B: Construct a Simple Centrifuge

6. Prepare the Salad Spinner

a. Gather your salad spinner and ensure it is clean and ready for use.

b. If you have small containers, fill them with water or a mixture (e.g., muddy water) to demonstrate separation.

7. Load the Salad Spinner

a. Place the filled containers securely inside the salad spinner. Make sure they are balanced; if using multiple containers, ensure they are evenly distributed.

8. Spin the Salad Spinner

a. Securely close the lid of the salad spinner.

b. Spin the salad spinner vigorously for about 1-2 minutes.

c. Observe what happens to the contents of the containers during spinning.

9. Observe Results

a. After spinning, open the salad spinner and examine the contents of each container.

b. Note any separation that has occurred (e.g., heavier particles settling at the bottom).

10. In your notes, create a table summarizing your findings:

a. Describe what was in each container before and after spinning.

b. Discuss how effective the salad spinner was in separating different components based on density.

Activity 5.19 Research and Present on Physics Concepts

Objective: Research a specific topic. You will prepare a presentation to share your findings with the class.

Materials Needed

1. Access to the internet (for research)

2. Books or articles on physics topics (optional)

3. Notebook and pen/pencil (for taking notes)

4. Presentation software (e.g., PowerPoint, Google Slides) or poster board (for creating your presentation)

5. Projector or display screen (if using digital presentations) What to do Organise yourselves into groups of no more than five.

1. As a group, select one of the following topics for your research:

a. The physics of banked roads

b. The use of centrifuges in medicine

c. Another application of circular motion of your choice!

2. Use the internet, library resources, and textbooks to gather information about your chosen topic. Focus on understanding the key concepts, principles, and applications. Take notes on important points, including:

a. Definitions and explanations of relevant physics concepts

b. Real-world applications and examples

c. Any interesting facts or recent developments related to your topic

3. Create an outline for your presentation. Organise your notes into sections that logically flow from one point to the next. Typical sections might include:

a. Introduction to the topic

b. Key principles and concepts

c. Applications in real life

d. Conclusion and summary of findings

4. Present your research to the class as a group. During your presentation, clearly explain each section of your topic.

5. After all presentations are completed, take some time to reflect on what you learned from both your research and from listening to others’ presentations. Write down any new insights or interesting facts you discovered during this activity

When a body is tied to a string and whirled in a vertical circle, the weight and the tension both contribute to the net centripetal force required to keep the body moving along its circular path.

At the top of the circle, the tension Tₜₒₚ and gravitational force mg both acts downwards. The centripetal force required to keep the object moving in a circle is provided by the sum of the tension and the weight:

Tₜₒₚ + mg = mv²____ r Tₜₒₚ = mv²____ r − mg The tension is minimum at the top of the circle.

∴ Tₘᵢₙ = mv²____ r − mg At the bottom of the circle, the tension T_(bottom) acts upward while the gravitational force mg acts downward. The centripetal force is provided by the difference between the tension and the weight:

T_(bottom) − mg = mv²____ r T_(bottom) = mv²____ r + mg The tension is maximum at the bottom of the circle.

∴ Tₘₐₓ = mv²____ r + mg

Activity 5.20 Watch videos about the design of loop-the-loops or playground swings Objective: Watch videos that explain the physics and design principles behind loop-the-loops in roller coasters and the energy transformations involved, focusing on concepts such as energy conservation and centripetal force.

Materials Needed

1. Access to a computer or tablet with internet access

2. Projector or large screen (if watching as a group)

3. Paper and pens for notes

4. Access to the video links:

a. Calculating the Minimum Height for a Perfect Ride

b. Ball Rolling Down an Inclined Track and Around a Loop What to do

1. Form small groups of not more than five. Each group will discuss their observations after watching the videos.

2. Open the first video link on your device or project it on a larger screen for the group. Watch the video titled “Calculating the Minimum Height for a Perfect Ride,” which explains how to calculate the minimum height needed for a sphere to complete a loop-the-loop without falling off.

3. While watching, take notes on:

a. Key concepts related to energy conservation and centripetal force in the context of loop-the-loops.

b. The calculations presented regarding height, speed, and forces involved.

c. Any specific design considerations mentioned that are important for ensuring safety and functionality.

4. After discussing the first video, open the second video link on your device or project it on a larger screen. Watch the video titled “Ball Rolling Down an Inclined Track and Around a Loop,” which demonstrates how potential energy is converted into kinetic energy as a ball rolls down an incline and around a loop.

5. While watching, take notes on:

a. the different forms of energy discussed (potential energy, kinetic energy).

b. how energy transformations affect the ball’s ability to complete the loop.

c. the relationship between initial height and velocity at different points in motion.

6. After watching both videos, come together in your groups to discuss your notes. Use the following prompts to guide your discussion:

a. What are the main forces acting on a roller coaster or ball at different points in a loop?

b. How does energy conservation play a role in designing safe loop- the-loops?

c. What factors can affect the design of a loop-the-loop (e.g., speed, height, radius)?

d. Can you think of real-life examples of roller coasters or similar systems that effectively use these principles?

7. Each group should prepare to share one key insight or interesting fact they learned from both videos with the class.

8. Discuss how these concepts apply to real-world scenarios and why they are significant in understanding physics.

Activity 5.21 Derive the mathematical relationship between the tension, mass, gravity, velocity and radius for the top and bottom of the circular path What to do

1. Organise yourselves into groups of no more than five.

2. Draw a diagram showing an object tied to a string with the body at the bottom of the circle such that the angle between the vertical and the string is zero.

3. Show the forces acting on the body (i.e. the weight, centripetal force and the tension) when the object is at the bottom of the circle.

4. Write down an equation that shows the relationship between the tension, the centripetal force and the weight

5. Make the tension the subject of the equation i.e. tension at the bottom

6. Show the forces acting on the body (i.e. the weight, centripetal force and the tension) when the object is at the top of the circle.

7. Write down an equation that shows the relationship between the tension, the centripetal force and the weight

8. Make the tension the subject of the equation i.e. tension at the top

9. From the two tension equations, take note of the maximum and minimum tension.

Activity 5.22 Investigating the tension for objects moving in vertical circles using a simulation Objective: Use an online simulation to explore how tension varies for an object moving in vertical circles, examining the effects of speed, radius, and position on tension.

Materials Needed

1. Access to a computer or tablet with internet access

2. Paper and pens for notes

3. Access to the simulation: Vertical Circle Simulation What to do

1. Organise yourselves into groups of no more than five. Each group will work together to explore the simulation.

2. Click on the provided link to access the Vertical Circle Simulation. Click on “Launch Interactive”. Once the simulation loads, familiarise yourself with the interface and available controls.

3. Start initial exploration with the default settings of the simulation. Take

note of the initial values for mass, radius, and speed displayed.

4. Investigate Tension at Different Positions

a. At the Top of the Circle

i. Observe the tension when the object is at the top of its vertical path.

ii. Record the value of tension and note how it relates to gravitational force.

b. At the Bottom of the Circle

i. Move the object to the bottom of its vertical path and observe the tension again.

ii. Record this value and compare it with the tension at the top.

c. At Intermediate Positions

i. Move the object to various points between the top and bottom (e.g., halfway up) and observe how tension changes.

ii. Take notes on how tension varies with position in relation to speed and gravitational force.

5. Adjust Variables

a. Change Mass

i. Adjust the mass of the object and observe how this affects tension at different positions.

ii. Record your observations regarding how mass influences tension.

b. Change Speed

i. Modify the speed of the object in circular motion.

ii. Note how increasing or decreasing speed affects tension at various points in the circle.

6. Create a table in your notes to summarise your findings:

a. Record different values of tension at various positions (top, bottom, intermediate) for different masses and speeds. Note any patterns or relationships observed between these variables.

7. After exploring various settings in the simulation, come together as a group to discuss your findings. Use these prompts to guide your discussion:

a. What patterns did you notice between position, mass, speed, and tension?

b. How does changing one variable affect others in terms of motion dynamics?

c. What practical applications can you think of that relate to vertical circular motion (e.g., amusement park rides, pendulums)?

Activity 5.23 Calculating Maximum and minimum tension in a string Study the worked examples carefully below before attempting the example questions that follow

Worked Example 1

A roller coaster car of mass 500 kg moves through a vertical loop of radius 10 m. At the top of the loop, the car has a speed of 12 m/s.

a. Calculate the tension in the track at the top of the loop.

b. If the car’s speed at the bottom of the loop is 18 m/s, calculate the tension in the track at the bottom.

Solution

Step 1 Identify the given variables v (top) = 12 m/s, v (bottom) = 118 m/s, m = 500 kg, r = 10 m, g = 9.8 m/s²Step 2 Identify the appropriate formula for tension at the top Tₜₒₚ = mv²____ r − mg

Step 3: Substitute the given values into the formula Tₜₒₚ = 500 × 12²_______ 10 − (500 × 9.8)

Step 4: Calculate to obtain the tension at the top Tₜₒₚ = 7200 − 4900 = 2300 N

Step 5: Identify the appropriate formula for tension at the bottom T_(bottom) = mv²____ r + mg

Step 6: Substitute the given values into the formula T_(bottom) = (500) × 15²_________ 10 + 500 × 9.8

Step 7: Calculate to obtain the tension at the bottom T_(bottom) = 11250 + 4900 = 16150 N Practice Problems Now, using the worked example as a guide, solve the following problems individually or in groups.

1. A 2-kg ball is attached to a string and whirled in a vertical circle of radius 1.5 m. At the bottom of the circle, the ball has a speed of 5 m/s. Calculate the tension in the string at the bottom of the circle. (g= 9.8 ms⁻²).

2. A 0.5-kg stone is tied to a string and swung in a vertical circle of radius 0.8 m. At the top of the circle, the stone moves at a speed of 3 m/s. Calculate the tension in the string at the top of the circle. (g= 9.8 ms⁻²).

Conical Pendulum

A conical pendulum is a type of pendulum in which the bob moves in a horizontal circular path while the string traces out a conical shape. This motion results from the combination of gravitational force, tension in the string, and the centripetal force required to maintain the circular motion.

Figure 5.9: Schematic diagram of Conical pendulum In the diagram above, body A has weight mg which acts vertically downward. The tension T in the string which makes an angle θ with the vertical has components T sin θ which is directed towards the and T cos θ which balances the weight mg of the body.

T cos θ = mg ------------------eqn (1) Since the component of the tension directed towards the centre is T sinθ T sin θ = m v²____ r , ------------------eqn (2), m v²____ r is the centripetal force eqn (2) ÷ eqn (1) T sinθ_____ T cos = m v²____ r ÷ mg tan θ = v²__ rg

Activity 5.24 Investigating conical pendulums using a simulation.

Objective: Use an online simulation to explore the dynamics of conical pendulums, examining how variables such as string length, and angular velocity affect the motion.

Materials Needed

1. Access to a computer or tablet with internet access

2. Paper and pens for notes

3. Access to the simulation: Conical Pendulum Simulation on GeoGebra What to do

1. Organise yourselves into groups of not more than five.

2. Click on the provided link to access the GeoGebra conical pendulum simulation. Once the simulation loads, familiarise yourself with the interface and available controls.

3. Start initial exploration with the default settings of the simulation. Take

note of the initial values for string length, mass, and angular velocity displayed.

4. Investigate Variables

a. Change String Length

i. Adjust the length of the string using the slider.

ii. Observe how changing the string length affects the angle of the pendulum and its circular motion.

iii. Take notes on your observations regarding how it influences radius and speed.

b. Change Angular Velocity

i. Adjust the angular velocity and observe how it impacts the angle of inclination and speed of circular motion.

ii. Take notes on how increasing or decreasing angular velocity affects centripetal force and motion stability.

5. Create a table in your notes to summarise your findings:

a. Record different combinations of string length, mass, angular velocity, angle of inclination, and any observed effects on motion.

b. Note any patterns or relationships you observe between these variables.

6. After exploring various settings in the simulation, come together as a group to discuss your findings. Use these prompts to guide your discussion:

a. What patterns did you notice between string length, mass, angular velocity, and motion?

b. How does changing one variable affect others in terms of motion dynamics?

c. What practical applications can you think of that relate to conical pendulums (e.g., amusement park rides, planetary motion)?

Review Questions 5.1

1. Name four examples of projectile motion in any area of life.

2. For a given initial velocity, State the factor that determines the maximum range of a projectile and state its numerical value.

3. Define the following terms:

a. Time of flight

b. Maximum height

c. Range of a projectile

4. A projectile is launched with an initial velocity of 40 m/s at an angle of 45 ° to the horizontal.

a. Find the horizontal range of the projectile.

b. Calculate the maximum height reached by the projectile.

5. You are in a debate with your colleagues about the optimal angle to kick a football to achieve the maximum horizontal distance. Your colleagues suggest angles of 30⁰, 50⁰and 70⁰. Given that the initial velocity of the football is 25 ms⁻¹and acceleration due to gravity g =10 ms⁻², use the principles of projectile motion to demonstrate that 45⁰is the optimal angle for achieving maximum range.

Review Questions 5.2

1. Define friction.

2. Name the two types of friction

3. Explain why it is necessary to lubricate moving parts of a machine with oil.

4. A girl can push a block of wood easily in a horizontal direction. When her brother decided to press vertically on the block, she could no longer move it as easily as before. Explain why this is so and what the girl needs to do to move the block.

5. A force of 3 N can move a bucket full of water horizontally on a tiled floor.

To prevent children from pushing the bucket easily, a woman decided to put a block on it. What is the minimum mass of a block needed for this purpose? [g=10ms², coefficient of static friction is 0.01, mass of bucket and water = 5 kg]

Review Questions 5.3

1. State two examples of circular motion

2. A car tyre with a radius of 0.5 m rotates at an angular velocity of 10 rad/s.

Determine:

a. The linear velocity of a point on the edge of the tyre.

b. The angular displacement of the tyre in 5 seconds.

3. A fan completes 120 revolutions in 1 minute. Calculate its angular velocity in radians per second.

Review Questions 5.4

1. At what point in the motion of an object moving in a vertical circular path (attached to a string) is the tension in the string at its maximum: at the top of the circle, at the bottom, or elsewhere? Explain your reasoning.

2. a. A curve on a road is banked at an angle of 10°, and a car is moving at a speed of 15 m/s. a. What is the radius of the curve for the car to move safely without friction?

b. Assuming the road is a four-lane road and the radius given measures up to the outer lane, do you think a car in the inner lane can safely negotiate the curve at the same speed as the car in the outer lane?

Explain.