Circular Motion (AQA A Level Physics)

A lead ball of mass 0.45 kg is swung round on the end of a string so that the ball moves in a horizontal circle of radius 1.3 m. The ball travels at a constant speed of 7.2 m s^–1. 

Calculate the time taken for the string to turn through an angle of 190º.

Calculate the tension in the string.           

You may assume that the string is horizontal.

The string will break when the tension exceeds a maximum tension. The ball makes two revolutions per second at the maximum tension of the string. 

Calculate the tension above which the string will break.

Discuss the motion of the ball in terms of the forces that act on it. In your answer you should: 

• Explain how Newton’s three laws of motion apply to its motion in a circle

• Explain why, in practice, the string will not be horizontal. 

You may wish to draw a diagram to clarify your answer.

The quality of your written communication will be assessed in your answer.

Explain why a particle moving in a circle with uniform speed is accelerating.

Figure 1 shows a fairground ride called a Rotor. Riders stand on a wooden floor and lean against the cylindrical wall. 

Figure 1

The fairground ride is then rotated. When the ride is rotating sufficiently quickly the wooden floor is lowered. The riders remain pinned to the wall by the effects of the motion. When the speed of rotation is reduced, the riders slide down the wall and land on the floor. 

At the instant shown in Figure 2 the ride is rotating quickly enough to hold a rider at a constant height when the floor has been lowered. 

Figure 2

Explain why the riders slide down the wall as the ride slows down.

A rotor accelerates uniformly from rest to maximum angular velocity of 3.6 rads s^–1. 

At the maximum speed the centripetal acceleration is 35 m s^–2. 

Calculate the diameter of the rotor.

Figure 3 shows the final section of a roller coaster which ends in a vertical loop. Cars on the roller coaster descend to the start of the loop and then travel around it. 

Figure 3

As the passengers move around the circle from A to B to C, the reaction force between exerted by their seat varies. 

State the position at which this force will be a maximum and the position at which it will be a minimum. Explain your answers.        

Figure 1 below shows a child at position B on a rotating playground roundabout. Frictional forces act on the child to keep them in the same position. 

Figure 1

The child is at a distance of 0.153 m from the centre of the centre of the roundabout. The linear speed of the child at B is 1.12 m s^–1. 

Calculate the revolutions per minute of the roundabout.

Annotate Figure 1 with an arrow to show the direction of the resultant horizontal force on the child.

The child has a mass of 20 kg. 

Calculate the centripetal acceleration and centripetal force at position B.

The roundabout has an inner railing to hold onto at its centre. There is another railing near the outer edge. 

Whilst the roundabout is rotating, state which railing would be easier to hold onto for the child and suggest why this would be so.

Figure 1 shows a ball of mass 160 g, which is fixed to the end of a light string of length 35 cm, is released from rest at X. It swings in a circular path, passing through the lowest point Y at speed v. 

Figure 1

Calculate the velocity of the mass at Y. 

Calculate the centripetal force acting on the mass.

Calculate the tension in the string.

The ball is replaced with a smaller ball of mass m attached to the same string and travels at a speed of 3.5 m s^–1 at point Y. The ball swings in a circular path identical to the one in Figure 1. 

If the tension of the string remains the same at point Y, calculate the mass m.

A proton of mass  is moving in a circle at constant speed. The time period of the particle’s rotation is T and its kinetic energy is E.

Show that the radius of the particle’s path is given by the formula: 

         r =

Figure 1 shows the path of the proton moving anti-clockwise in a circle at constant speed. The proton travels 1.24 mm during one full revolution. 

Figure 1

The net force acting on the proton is 64 fN.  

Show that the speed of the proton is 8.7 × 10⁴ ms^–1.

Calculate the frequency of rotation of the proton.

Explain why the work done by the centripetal force on the proton is zero.

A small ball is attached to a string and moves in a circle with constant angular speed, . The angle between the string and the vertical is θ, as shown in Figure 1.  

Figure 1

Show that the radius of the ball’s path can be given by the formula: 

         r  =

The ball has a mass of 413 g and the angle between the string and the vertical θ is 19°. It rotates with an angular speed of 3.42 rad s^-1. 

Calculate the radius of the ball’s path. 

Give your answer to an appropriate number of significant figures.

Determine the number of oscillations per minute that the ball completes, assuming its angular speed remains constant.

Explain the effect upon the oscillations per minute of extending the length of the string, whilst keeping the radius of the circular motion, r, constant. 

Assume the mass of the string is negligible.

A satellite orbits the Earth, as shown in Figure 1. It remains in orbit due to the gravitational field of the Earth.

Figure 1

It takes the satellite 100 minutes to complete one full revolution around the Earth. The mass of the Earth is 6.0×10²⁴ kg and the radius of the Earth is 6400 km. 

Show that the distance of the space telescope from the surface of the Earth is approximately 750 km.

A manned space station orbits at a distance of 1500 km from the surface of the Earth. 

Calculate the orbital period of the space station.

During an emergency, the space station in part (b) needs to rendezvous with the satellite in part (a). 

Determine the increase in the orbital speed of the space station as moves from orbiting at its original height to the height of the satellite

The space station is required to tether itself to the satellite. This requires the two to be joined by a 20 m cord, as shown in Figure 2. 

Figure 2

Explain why the arrangement shown in Figure 2 will require the space station to use some of its fuel whilst tethered at this height if it wishes to maintain the original orbital period of the satellite. 

A roller coaster cart is travelling over a section of track, shown in Figure 1, at a constant speed, v. N passengers have an average mass,  , and the cart has a mass, . An electric motor on the cart is used so it can maintain its speed as it climbs, and then goes over a bump in the track, that has a radius of the curvature equal to r. 

Figure 1

Determine a general formula for the amount of electrical energy, , that needs to be used to maintain the speed v, over the bump.

The radius of the curvature of the path of the passengers is 21 m, and the cart maintains a speed of 8 m s^-1. The average mass of the passenger, ,  is 56 kg.

If 5 passengers ride the cart over the bump together, estimate the change in the contact force, R, between the passenger and the seat of the cart as the car travels over the highest point of the bump.

A banked turn is a section of road that is angled so that cars can travel around corners at a higher velocity without the aid of friction. 

Several racetracks have banked turns that are curved, such that the angle of the track increases as the distance from the inside edge of the track increases. An image of such a track is shown in Figure 2. 

Figure 2

Discuss the motion of the car as it travels around the banked turn. In your answer you should: 

• Explain how Newton’s three laws of motion apply to its motion in a circle

• Explain the effect of changing the angle of the road, and the speed of the car, on its motion 

You may wish to draw a diagram to clarify your answer.

The quality of your written communication will be assessed in your answer.