6.1 Circular Motion

A small ball is attached to a string and moves in a horizontal circular path. It completes one revolution every 2.5 s, with the string at an angle θ to the vertical.

Calculate the orbital radius r if θ = 12°. 

You may wish to use the following data: 

Show that the length of the string l is given by: 

You may wish to use the following data: 

The equation in part (b) seems to suggest that the length of the string l is dependent on the angle it makes to the vertical, θ.

Comment on the relationship between the length of the string l and the angle it makes to the vertical, θ. 

A marble rolls from the top of a bowling ball of radius R. 

Show that when the marble has moved so that the line joining it to the centre of the sphere subtends an angle of θ to the vertical, its speed v is given by: 

Deduce that, at the instant shown in the image in part (a), the normal reaction force N on the marble from the bowling ball is given by:

Hence, determine the angle θ at which the marble loses contact with the bowling ball. 

The 'loop-the-loop' is a popular ride at amusement parks, involving passengers in cars travelling in a vertical circle. 

The loop has a radius of 8.0 m and a passenger of mass 70 kg travels at 10 m s^–1 when at the highest point of the loop. 

Calculate, at the highest point:

Stating any assumptions required, calculate the speed of the passenger at the point marked 'exit from loop' in part (a). 

Operators must ensure that the speed of the vehicle carrying passengers into the loop-the-loop is above a certain value. 

Suggest a reason for this, and determine the minimum required speed. 

A popular trick to impress young observers is to swing a bucket of water in a vertical circle. If the bucket is swung fast enough, no water spills out. 

Estimate the minimum linear speed v required to swing a bucket in a vertical circle, such that no water spills. 

When the bucket of water is stirred with a spoon in uniform circular motion near the rim, the level of water in the bucket is observed to change from a flat horizontal dashed line to a curved solid line, as shown.

By considering the circular motion of a fluid particle in the water, explain this observation using relevant physical principles. 

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

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

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 below the ride is rotating quickly enough to hold a rider at a constant height when the floor has been lowered.

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.

The diagram 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.

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. 

A centrifuge is often used in astronaut training. This is to simulate Earth’s gravity on board the space station. The astronauts sit in a cockpit at the end of each arm, each rotating about an axis at the centre.

At its top speed, the centrifuge makes 1 full rotation every 2.30 s.

Calculate the frequency of the centrifuge. State an appropriate unit and express your answer to an appropriate number of significant figures.

Calculate the angular speed of the centrifuge in rad s^–1.

Each astronaut is placed 6.30 m from the rotation axis.

Calculate the magnitude of the centripetal acceleration on each astronaut.

Sketch the direction of the acceleration on each astronaut.

A section of a roller coaster carries a passenger over a curve in the track. The radius of curvature of the path of the passenger is r and the roller coaster is travelling at constant speed v. The mass of the passenger is m.

At a particular point on the track, the car moves with a linear velocity of 22 m s^–1. The reaction force between the car and the track at this point is 210 N and the passenger has a mass of 65 kg.

Calculate the distance from the passenger to the centre of curvature of the curved track.

State and explain what would happen to the magnitude of R if the rollercoaster passed over the curved track at a higher speed.

When the rollercoaster passes over a curved section of a track above a certain speed, the passenger is momentarily lifted off their seat and experiences weightlessness.

Calculate the speed at which the rollercoaster must be travelling for the passenger to experience weightlessness.

The London Eye shown in the diagram has a radius of approximately 68 m and the passengers in the capsules travel at an angular speed of 3.5 × 10^–3 rad s^–1.

Calculate the speed of each passenger in the capsules.

Assume the London Eye is rotating clockwise.

Sketch the following on any capsule:

Each capsule weighs about 98.1 kN.

Calculate the centripetal force on an empty capsule.

Dan has travelled to London to watch an exciting Physics show. Being an eager tourist, he arrives early and plans to ride the London Eye. When he gets to the front of the queue however, he realises he only had 40 minutes before he needs to leave for the show.

State, with a calculation, whether Dan is still able to ride the London Eye and leave to see the show on time.