Centripetal Acceleration (CIE A Level Physics)

Two runners are running around a circular track. One runner follows path X and the other follows path Y, as shown in Fig. 1.1.

The radius of path X is 36.50 m. Path Y is parallel to, and 1.2 m outside, path X. Both runners have mass 75 kg. The maximum lateral (sideways) friction force F that the runners can experience without sliding is 57 mN. 

Show that the maximum speed at which the runner on path Y can move is 0.17 m s^–1.

Compare the centripetal acceleration and maximum speed of the runner on path X compared to path Y.

Calculate the time taken for the runner in path X to complete 1 lap.

The runner on path Y has an ankle injury that has come back with a vengeance, so it takes them twice as long to complete 1 lap.

A bowl-shaped skating ramp can be modelled as a hollow sphere.

A skateboarder with a horizontal speed follows a circular path inside the bowl of radius 2.3 m, as shown in Fig 1.1.

Fig 1.1

The forces acting on the skateboarder are their weight W and the normal reaction force R of the ramp on the skateboarder. R is at an angle θ to the horizontal. 

Determine an equation for W in terms of θ and the resultant force, F on the skateboarder.

Explain why force F is significant to the motion of the skateboarder.

For the radius of the ramp in Fig 1.1, the skateboarder travels at a speed of 6.0 m s^–1. 

Show that the angle θ is 32°.                          

Fig. 1.1 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. 

Fig 1.1

The child is at a distance of 0.167 m from the centre of the roundabout. The linear speed of the child at B is 1.23 m s^–1. 

Calculate the number of times the roundabout revolves in one minute.

Mark an arrow on Fig. 1.1 to show the direction of the resultant horizontal force on the child.

The child has a mass of 23.4 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.

A ‘loop the loop’ is a vertical circular path. Fig. 1.1 shows a toy car of mass 70 g completing a ‘loop the loop’ which has a radius of 0.15 m. 

Fig. 1.1

The car enters the loop with a speed of 2.0 m s^–1 and then undergoes a constant deceleration until it passes A with a speed of 1.21 m s^–1. 

After A, the car undergoes a constant acceleration before leaving the loop with a speed of 2.0 m s^–1 

For the journey through the loop, sketch a graph to show how the reaction force on the car varies with the angle rotated by the car around the loop. 

Include the values for the maximum and minimum forces on the car. 

Only show the general shape between these values.

A student is investigating the energy changes in a ‘loop the loop’, as shown in Fig. 1.2.

They want to determine the minimum height, h, required for a toy car of mass m to complete a loop the loop of radius r, when starting from rest. 

Fig. 1.2

Show that the minimum height, h is:

A satellite orbits the Earth, as shown in Fig. 1.1. It remains in orbit due to the gravitational field of the Earth.

Fig. 1.1

It takes the satellite 200 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 satellite from the surface of the Earth is approximately 5000 km.

Calculate the acceleration due to the Earth's gravity of the satellite at its orbiting altitude from part a). 

The satellite crashes into the surface of the Earth and becomes embedded in the ground. 

Determine a ratio for the angular speed of the satellite on the surface of the Earth and the angular speed of the satellite when it was in orbit in part a). Explain your answer.