Circular motion

The diagram shows a section through the components of a speedometer for a bicycle. The front wheel axle of the bicycle is connected to a cable. The cable rotates as the bicycle wheel rotates. The other end of the cable is connected to a magnet which also rotates, as shown.

An aluminium disc is shaped so that it is close to, but not touching, the magnet. The disc is connected to a pointer by a shaft. A spring prevents the disc and shaft from freely rotating.

*(a) Explain why the aluminium disc rotates in the same direction as the magnet.

The road speed of the bicycle should be displayed by the speedometer.

Explain why the displayed speed can not be accurate, as bicycles can have different wheel diameters.

A roundabout completes 5.0 revolutions in 20 s.

Which of the following is the angular velocity of the roundabout?

Regenerative braking supplies a current back to the power transmission system whilst slowing a vehicle. The arrangement shown can be used as a regenerative braking system on a train.

The coil rotates with the wheels of the train. Two copper rings are connected to the ends of the coil. The rings rotate with the coil and two carbon blocks make electrical contact with the rings as they rotate.

Describe how this arrangement can be used as a regenerative brake.

A specification for a new train states that the train should be able to accelerate to a speed of 360 km hour⁻¹ from rest, and that this acceleration should be completed within 40 km of level track.

The graph shows the performance of the train on a test run.

Calculate the acceleration of the train as it accelerates to a speed of

Deduce whether the performance of the train met the specification on this test run.

On curved tracks there is a maximum safe speed for the train.

Explain why there is a maximum safe speed for a train travelling on a curved track.

When the train travels at 200 km hour⁻¹, the minimum safe radius of curvature of the track is 1800 m.

Calculate the minimum safe radius of curvature for a speed of 360 km hour⁻¹.

The plan view of a model racing car track is shown. Friction acts between a model racing car and the track. A car is moving round the track with a constant speed and reaches point P. Arrows indicating directions North and South are also shown.

The car then slides off the track at P.

Which of the following is the reason why the car slides off the track?

A ‘tennis trainer’ consists of a tennis ball suspended by a string from the top of a vertical pole. When the ball is hit it travels in a horizontal circle around the pole, as shown in both the photograph and the diagram.

The radius of the path of the ball is 1.2 m and the speed of the ball is 3.8 m s⁻¹.

Deduce whether these values are consistent with the angle between the string and the vertical pole shown in the photograph.

A device called a clutch can be used to connect a motor to a load. The diagram shows a design called an eddy current clutch.

Several magnets are embedded in the plastic disc and it is rotated by the motor.

Explain why a current is induced in the copper disc when the motor is switched on.

Explain, using Lenz’s law, why the copper disc rotates.

The motor rotates at 500 revolutions per minute.

Calculate the angular speed ω of the motor.

The table shows how the turning effect exerted on a load varies with ω for a particular distance between the copper disc and the plastic disc.

Explain the trend shown by the data.

The International Space Station (ISS) orbits the Earth with a constant speed v. The orbit is circular and of radius r.

The diagram represents two positions, A and B, of ISS during its orbit.

Draw a labelled vector diagram, in the space below, of the velocities at the two positions that shows the acceleration is directed towards the centre of the orbit.

The ISS completes one orbit in 92 minutes.

Calculate the centripetal acceleration of the ISS.

r = 6800 km

Astronauts in the ISS are often described as being “weightless”.

Discuss whether the astronauts are “weightless” when they are orbiting the Earth in the ISS.

The Enterprise is an amusement park ride. Riders sit in cars that are made to rotate in a vertical circle.

The ride starts by moving in a horizontal circle. The speed of rotation increases, and the frame tilts until the ride is rotating vertically as shown.

The photograph below shows riders at the top of the vertical circle. The riders are in contact with their seats at all times during the ride.

The diagram shows the weight W of a rider and the push P from the seat on the rider at the top and bottom of the circular path.

*(a) The rider moves from the bottom to the top of the circular path.

Explain how the apparent weight experienced by the rider would change.

On the website of the amusement park it states

“The ride is perfectly safe without the need for safety harnesses for the riders. Centrifugal force ensures that the riders remain in their seats at all stages in the ride.”

Assess the validity of this statement.

The blade of a lawnmower rotates at a speed of 50 revolutions per second.

Which of the following is the angular speed of the blade in rads s⁻¹?

The blade of a lawnmower rotates at a speed of 50 revolutions per second.

Which of the following is the angular speed of the blade in rads s⁻¹?

The photograph shows a model racing car set. The curved parts of the track are semicircular. The car makes electrical contact with the track using metal brushes underneath the car.

There is a maximum speed for the car to stay on the curved part of the track. Explain why the car will slip off the curved part of the track if the car exceeds the maximum speed.

The following measurements are made for a car starting at rest on a straight piece of track.

distance travelled = 1.2 m

time taken = 0.77 s

Show that the final velocity of the car is about 3 m s⁻¹.

Assume the acceleration is constant.

The final velocity calculated in (b)(i) is the maximum velocity before the car slips off the track.

Calculate the maximum horizontal force between the curved part of the track and the car.

mass of car = 0.050 kg

radius of curved part of track = 0.042 m

The cars are controlled separately and so can be raced, with one car on the inner lane and the other on the outer lane. The cars are identical. Each car is raced at its highest speed for that lane.

Explain why the outcome of the race is difficult to predict.

The photograph shows a model racing car set. The curved parts of the track are semicircular. The car makes electrical contact with the track using metal brushes underneath the car.

There is a maximum speed for the car to stay on the curved part of the track. Explain why the car will slip off the curved part of the track if the car exceeds the maximum speed.

The following measurements are made for a car starting at rest on a straight piece of track.

distance travelled = 1.2 m

time taken = 0.77 s

Show that the final velocity of the car is about 3 m s⁻¹.

Assume the acceleration is constant.

The final velocity calculated in (b)(i) is the maximum velocity before the car slips off the track.

Calculate the maximum horizontal force between the curved part of the track and the car.

mass of car = 0.050 kg

radius of curved part of track = 0.042 m

The cars are controlled separately and so can be raced, with one car on the inner lane and the other on the outer lane. The cars are identical. Each car is raced at its highest speed for that lane.

Explain why the outcome of the race is difficult to predict.

Scientists have been studying a type of jumping spider that can jump up to six times its body length.

The scientists photographed a spider at 0.02s intervals, during a jump. The picture is taken from the photograph and is shown actual size.

Deduce whether the images show that the motion in the x-direction is independent of the motion in the y-direction. You should take measurements using the cross marking the centre of gravity of the spider.

Show that the initial velocity of the spider at the start of the jump is about 1 m s⁻¹. You should take measurements using the cross marking the centre of gravity of the spider.

The spider achieves this jump by extending its two back legs by 3.0 mm.

Calculate the average force the spider exerts in each leg to achieve the jump.

mass of spider = 150 mg

Just as the spider starts the jump, it fixes a silk thread to the platform. It is thought that the thread acts as a safety line in case the spider falls.

A student makes the comment:

ʻIf the silk thread can withstand a tension equal to the weight of the spider then this safety system should work.ʼ

Deduce whether this statement is correct.

At the end of the 19th century, J.J. Thompson used electric and magnetic fields to deflect beams of charged particles. A photograph of his apparatus is shown.

Electrons were accelerated through a potential difference to produce a beam of high-energy electrons. The beam was then deflected in perpendicular directions by the magnetic and electric fields. The final position of the beam on the screen was determined by the charge and mass of the electrons.

Explain how electrons from the source become a beam of high-energy electrons.

An electron is travelling left to right and enters a region of uniform magnetic field as shown below. The direction of the magnetic field is perpendicular to the direction of travel of the electron.

The magnetic field deflects the electron in the direction up the page.

Explain the direction of the magnetic field that would produce this deflection.

Explain why the electron would travel in a circular path if no other forces acted on it.

In a modern version of Thompson’s experiment, a uniform electric field of electric field strength E is applied so that the electric and magnetic forces on the electrons are equal and in opposite directions.

Show that for electrons to be undeflected their velocity must be given by

where B is the magnetic flux density of the magnetic field.

The beam is produced by accelerating electrons through a potential difference of 250 V. The electric field strength is 1.4 × 10⁴V m⁻¹. The magnetic flux density is

Calculate the value of the specific charge e/m for the electron using this data.

In his original experiments, Thompson determined the specific charge of a range of particles. His results indicated that the specific charge of an electron is about 2000 times bigger than that for a hydrogen ion.

Deduce what conclusion can be made from this information.

At the end of the 19th century, J.J. Thompson used electric and magnetic fields to deflect beams of charged particles. A photograph of his apparatus is shown.

Electrons were accelerated through a potential difference to produce a beam of high-energy electrons. The beam was then deflected in perpendicular directions by the magnetic and electric fields. The final position of the beam on the screen was determined by the charge and mass of the electrons.

Explain how electrons from the source become a beam of high-energy electrons.

An electron is travelling left to right and enters a region of uniform magnetic field as shown below. The direction of the magnetic field is perpendicular to the direction of travel of the electron.

The magnetic field deflects the electron in the direction up the page.

Explain the direction of the magnetic field that would produce this deflection.

Explain why the electron would travel in a circular path if no other forces acted on it.

In a modern version of Thompson’s experiment, a uniform electric field of electric field strength E is applied so that the electric and magnetic forces on the electrons are equal and in opposite directions.

Show that for electrons to be undeflected their velocity must be given by

where B is the magnetic flux density of the magnetic field.

The beam is produced by accelerating electrons through a potential difference of 250 V. The electric field strength is 1.4 × 10⁴V m⁻¹. The magnetic flux density is

Calculate the value of the specific charge e/m for the electron using this data.

In his original experiments, Thompson determined the specific charge of a range of particles. His results indicated that the specific charge of an electron is about 2000 times bigger than that for a hydrogen ion.

Deduce what conclusion can be made from this information.

A student was investigating the forces involved in circular motion.

He placed a small coin on a horizontal turntable as shown. The turntable was connected to a driver unit so that it could be rotated at a constant rate.

The student switched on the driver unit and increased the rate of rotation until the coin slid off the turntable. He read the angular velocity of the turntable from a digital display on the driver unit. He then replaced the coin in the original position on the turntable and repeated the procedure.

His results are shown.

The student used the results to calculate a mean value of .

State the purpose of calculating a mean.

Calculate the percentage uncertainty in the mean value of .

The student used and r to calculate the centripetal acceleration of the coin at the instant it started to slide.

Calculate the percentage uncertainty in this centripetal acceleration.

The student repeated the procedure with different values of r.

Explain how the value of at which the coin started to slide varied as r increased.

A centrifuge is a machine which rotates.

A particle in a centrifuge moves in a circle of radius r, centre O, with a constant speed v.

The diagram represents two positions of the particle.

Derive the equation for centripetal acceleration a = by considering the velocity at these two positions.

Your answer should include a vector diagram.

The United States’ space agency, NASA, uses a centrifuge to test whether equipment will operate when experiencing large forces. The equipment to be tested is attached to the end of the frame of the centrifuge, which rotates around a vertical axis at its centre.

The centrifuge rotates at 50 revolutions per minute with a radius of 8.8 m.

Show that the angular velocity of the centrifuge is about 5 rad s^–1.

Explain how the centrifuge applies large forces to the equipment under test.

The NASA website says the centrifuge can be used to test whether the equipment can withstand accelerations of up to about 25 g.

Deduce whether this claim is correct.

Motors usually have a rotating component which can do work W.

In the diagram below, electric field lines have been drawn around one pair of these spheres.

Add to the diagram to show • the directions of the field lines • the lines of equipotential.

The distance between the centres of each charged sphere in this pair is 5.0 cm.

Show that the force between this pair of charged spheres is about 0.04 N.

charge on each sphere = 0.10 μC

The table shows the typical power and the corresponding angular velocity required for three different appliances.

Deduce which of these appliances, in principle, could use the electrostatic motor in (b).

You should use the word equation in (a) and assume that the length of the rod in the electrostatic motor is 8.0 cm.

Assume that the electrostatic motor would deliver a constant force throughout one complete rotation.

Mass spectrometry is a technique used to separate ions based on their charge to mass ratio.

The atoms in a sample are ionised and then accelerated and formed into a fine beam. This beam is passed into a region of uniform magnetic field and the ions are deflected by different amounts according to their mass.

Analysis of mass spectrometer data shows that chlorine exists in nature as two isotopes, chlorine-35 and chlorine-37.

State what is meant by isotopes.

In a mass spectrometer, chlorine-35 ions are accelerated through a potential difference of 8.50 kV to produce an ion beam.

Show that the speed of singly ionised chlorine-35 atoms is about 2.2 × 10⁵ m s⁻¹.

mass of an ion of chlorine-35 = 34.97 u

In most mass spectrometers the ions are passed through a velocity selector, after being accelerated, to produce a beam of ions of a particular velocity. The velocity selector consists of a pair of parallel plates, across which a potential difference (p.d.) is applied to create an electric field.

In one mass spectrometer the plates are 2.5 cm apart and a p.d. of 135 V is applied.

A magnetic field is also applied to produce a force on the ions in the opposite direction to the force from the electric field. For one particular speed the ions travel in a straight line and emerge from the selector.

Add to the diagram to indicate the directions of the electric field and the magnetic field.

The magnetic flux density applied to the velocity selector is 24.5 mT.

Deduce whether this magnetic flux density is suitable to produce a beam of chlorine-35 ions of speed 2.2 × 10⁵ m s⁻¹.

After passing through the velocity selector the ion beam enters a region of uniform magnetic flux density 0.35 T with the ions travelling at right angles to the field direction.

Explain why the ions travel in a circular path.

Calculate the radius of the circular path.

The diagram shows the path of the chlorine-35 ions in the field. Chlorine-37 ions enter the magnetic field with the same velocity.

1. Add another line to the diagram to show the path of these chlorine-37 ions.

2. Explain any differences in the paths.