Circular motion

A racing car of mass m travels around a horizontal oval track. The curved sections of the track are semicircles.

At the instant shown in Figure 10 the car is moving with constant speed v in a circle of radius R.

The car has kinetic energy E_k. The resultant force acting on the car is F.

The maximum centripetal force that can be produced between the car’s tyres and the track is 24 kN. The minimum value of R is 230 m.

m = 1600 kg

The car just starts to slip when it travels on the track in a circular path of radius 230 m

Deduce the speed of the car.

The driver wishes to drive this car around the curved section of the track, without slipping, at a speed greater than your answer to Question 07.2.

Suggest one way in which the driver can achieve this.

Figure 11 shows the car on the curved section of a different oval track. The curved section of this track is a slope. This means that cars can travel at greater speeds than on the track shown in Figure 10.

The car in Figure 11 stays at the same height on the curved section of the track. There is no tendency for the car to slip up or down the track and therefore there is no sideways friction on the tyres.

Draw and label, on Figure 11, arrows to show the direction of each force that acts on the car as it travels around the curved section of the track.

A car has a greater maximum speed on a sloped track than on a horizontal track of the same radius.

Explain why.

A small mass is placed at P on a horizontal turntable. The turntable rotates clockwise with a constant angular speed about a vertical axis through its centre O.

The mass remains at rest relative to the turntable.

What is the direction of the frictional force on the mass at the instant shown?

A particle of mass m moves in a circle of radius r. The number of revolutions completed per second is f.

What is the kinetic energy of the particle?

Two circular discs made of card rotate at constant speed on a common axle.

The discs are 2.00 m apart.

An air-gun pellet is fired parallel to the axle. The pellet makes holes in the discs. The holes are separated by an angle of 45°. The speed of the pellet between the discs is 300 m s⁻¹.

How many revolutions does each disc complete in one second?

When an aircraft turns in a horizontal circular path, it banks at an angle θ.

The aircraft has mass m and travels at constant speed v in a horizontal circular path of radius r. The lift force acts at the angle θ.

What is tan θ ?

A teacher sets up a demonstration to show the relationship between circular motion and simple harmonic motion (SHM). She places a block on a turntable at a point 0.25 m from its centre, as shown in Figure 9.

The turntable rotates with an angular speed of 1.8 rad s^–1 and the block does not slip.

Calculate the time taken for the turntable to complete one revolution.

Figure 10 shows a plan view of the turntable and block. The turntable rotates in a clockwise direction.

Draw an arrow on Figure 10 to show the direction of the resultant force on the block.

The mass of the block is 0.12 kg.

Calculate the magnitude of the resultant force on the block.

Describe, with reference to one of Newton’s laws of motion, the evidence that a resultant force is acting on the block.

The teacher adjusts the angular speed of the turntable so that the block completes one rotation every 2.50 s. She sets up a simple pendulum above the centre of the turntable so that it swings in phase with the movement of the block.

Calculate the length of the simple pendulum.

A lamp is used to project shadow images of the block and pendulum bob on a screen. Both shadows appear to move with SHM across the screen. Figure 11 shows the images on the screen at one instant.

Initially the shadows move in phase with the same amplitude.

Air resistance affects the motion of the pendulum.

Suggest the effect this has on the amplitude relationship and the phase relationship between the moving shadows.

Figure 6 shows a rotating spacecraft that is proposed to carry astronauts to Mars.

The spacecraft consists of two parts A and B connected by a rigid cylindrical rod. When the spacecraft is travelling, A and B rotate at a constant angular speed about their common centre of mass O.

L is the distance between the centre of mass of A and the centre of mass of B. r_A is the distance from O to the centre of mass of A.

As the spacecraft rotates, a force that imitates the effect of gravity acts on an astronaut who is in contact with the floor.

Explain why.

The forces exerted on A and B by the connecting rod have the same magnitude.

Show, by considering the centripetal forces acting on A and B, that r_A is given by

In this spacecraft m_A < m_B.

Deduce whether the centre of mass of A or the centre of mass of B rotates with a greater linear speed.

The astronauts live in A and the cargo is stored in B.

When loaded,

The spacecraft imitates the gravity of Mars where g = 3.7 m s⁻².

Figure 7 shows a stress–strain curve for the metal used for the rigid rod.

Suggest a suitable diameter for the rod. Justify your answer.

A pair of cameras is used on a motorway to help determine the average speed of vehicles travelling between the two cameras.

Figure 5 shows the speed–time graph for a car moving between the two cameras.

The speed limit for the motorway between the two cameras is 22 m s⁻¹.

Determine whether the average speed of the car exceeded this speed limit.

Markings called chevrons are used on motorways.

The chevron separation is designed to give a driver time to respond to any change in speed of the car in front. The driver is advised to keep a minimum distance d behind the car in front, as shown in Figure 6.

Government research suggests that the typical time for a driver to respond is between 1.6 s and 2.0 s.

Suggest a value for d where the speed limit is 31 m s⁻¹.

The chevron separation is based on the response time, not on the time taken for a car to stop.

The brakes of a car are applied when its speed is 31 m s⁻¹ and the car comes to rest. The total mass of the car is 1200 kg.

The average braking force acting on the car is 6.8 kN.

Calculate the time taken for the braking force to stop the car and the distance travelled by the car in this time.

Suggest why the chevron separation on motorways does not take into account the distance travelled as a car comes to rest after the brakes are applied.

At bends on motorways the road is sloped so that a car is less likely to slide out of its lane when travelling at a high speed.

Figure 7 shows a car of mass 1200 kg travelling around a curve of radius 200 m. The motorway is sloped at an angle of 5.0°.

Figure 8 shows the weight W and reaction force N acting on the car. The advisory speed for the bend is chosen so that the friction force down the slope is zero.

Suggest an appropriate advisory speed for this section of the motorway.

An object of mass m moves in a circle of radius r. It completes n revolutions every second.

What is the kinetic energy of the object?

Figure 8 shows a side view of an act performed by two acrobats. Figure 9 shows the view from above.

The acrobats, each of mass 85 kg, are suspended from ropes attached to opposite edges of a circular platform that is at the top of a vertical pole. The platform has a diameter of 2.0 m A motor rotates the platform so that the acrobats move at a constant speed in a horizontal circle, on opposite sides of the pole.

When the period of rotation of the platform is 5.2 s, the centre of mass of each acrobat is 5.0 m below the platform and the ropes are at an angle of 28.5° to the vertical as shown in Figure 8.

Show that the linear speed of the acrobats is about 4.5 m s⁻¹

Determine the tension in each rope that supports the acrobats.

Discuss the consequences for the forces acting on the pole when one acrobat has a much greater mass than the other.

What is the tension in the string when the bob is at the bottom of the circle?