Circular motion

A washing machine with a circular drum of diameter 0.60 m is rotating at an angular velocity of 15 radians per second.

What is the magnitude of the force exerted by the drum on a 0.30 kg item of clothing which is at the bottom of the spinning drum?

A 21 N

B 23 N

C 41 N

D 43 N

Gravitational waves are disturbances in space-time that are produced by violent astrophysical events.

Define strain.

The maximum strain shown on the graph is approximately 10^–21. (ii)

Calculate the force that would produce the same strain in a copper wire of diameter 2 mm.

Young modulus of copper = 130 GPa

By considering the answer to (ii), suggest why it is very difficult to detect the strain caused by gravitational waves.

In this part of the question you will use Newtonian physics to estimate the power output of the black hole collision.

The diagram shows two identical black holes, each of mass M, orbiting each other with period T.

Prior to colliding, the black holes move at speed V in identical circular orbits of diameter D about the central point C.

Write down an expression for the gravitational force, F, acting between the black holes in terms of M, D and the universal gravitational constant, G.

The total orbital energy, E, of the black holes is the sum of their kinetic energy and gravitational potential energy. Show that E is given by the expression:

E = k (²G²M⁵)^1/3 (1/T^2/3)

where k is a numerical constant.

By considering times B and D, use your answer to (iii) to estimate the power radiated as gravitational waves in this time interval.

Suggest why the magnetic force acting on the proton has a different magnitude at X than at Y

At X, the motion of the proton is instantaneously equivalent to motion in a circle at a constant speed.

Calculate the radius of this circular motion.

1 Calculate the magnitude of the resultant force on the proton at Y.

2 Explain why the motion of the proton at Y is not instantaneously equivalent to motion in a circle at a constant speed.

A satellite is in geostationary orbit 36 000 km above the Earth’s surface. The Earth has a radius of 6400 km.

At what speed is the satellite moving relative to the centre of the Earth?

0 m s^–1 A

490 m s^–1 B

2.6 km s^–1 C

3.1 km s^–1 D

Your answer

Draw an arrow labelled T on the diagram to represent the tension in the rod.

Show that the radius of the circular path followed by the sandbag is about 10 m.

Calculate the tension T in the rod.

Show that the angular velocity of the ride is about 0.8 radians per second.

When the seat is at its highest point the sandbag is 17 m above the ground. The sandbag is released from the seat to model an object being dropped by a rider.

Calculate t, the time taken for the sandbag to reach the ground.

Using your answer to (a)(iv), determine the horizontal displacement s travelled by the sandbag before hitting the ground.

Determine, with reasons, the effect on the horizontal displacement travelled if the object released from the ride was a shoe from a rider.

The table shows some data on the planet Venus.

Calculate the magnitude of the gravitational field strength g at the surface of Venus.

Give your answer to 3 significant figures.

=........ N kg^–1 g

Two identical space probes, A and B, land on a flat surface on Venus.

Probe A lands at the north pole. Probe B lands on the equator.

Each probe has mass 760 kg and volume 1.7 m³.

Calculate the centripetal acceleration a of probe B at the equator due to the rotation of Venus about its axis.

=........ m s^–2 a

The atmosphere exerts the same upthrust on each probe.

Using your answer to (a), calculate the upthrust acting on each probe.

Explain which probe will experience the greater normal contact force from the surface of Venus.

A small object of mass m is placed on a rotating horizontal metal disc at a distance r from the centre of the disc.

The frequency of rotation is adjusted using a motor attached to the disc.

The frequency of rotation of the disc is slowly increased from zero, until the object slips off. At this point, the friction F acting on the object is equal to the centripetal force.

The friction F is given by the expression F = kmg, where k is a constant and g is the acceleration of free fall. The constant k has no units.

Both the ISS and the astronauts inside it are in free fall.

Explain why this makes the astronauts feel weightless.

Calculate the value of the gravitational field strength g at the height of the ISS above the Earth.

=........ N kg^–1 g

Use the information in (b)(ii) and the data below to show that the root mean square (r.m.s.) speed of the air molecules inside the ISS is approximately 15 times smaller than the orbital speed of the ISS.

• molar mass of air = 2.9 × 10^–2 kg mol^–1

• temperature of air inside the ISS = 20 °C

The ISS has arrays of solar cells on its wings. These solar cells charge batteries which power the ISS. The wings always face the Sun.

Use the data below and your answer to (b)(ii) to calculate the average power delivered to the batteries.

• The total area of the cells facing the solar radiation is 2500 m².

• 7% of the energy of the sunlight incident on the cells is stored in the batteries.

• The intensity of solar radiation at the orbit of the ISS is 1.4 kW m^–2 outside of the Earth’s shadow and zero inside it.

• The ISS passes through the Earth’s shadow for 35 minutes during each orbit.

At an airport, the conveyor belt for suitcases moves at a constant speed of 1.5 m s^–1. In Fig. 4.1, a suitcase of mass 8.0 kg has reached the line labelled XX'.

Fig. 4.2 shows the situation in vertical cross-section. The frictional force F prevents the suitcase of weight W from sliding to the bottom of the belt. The normal contact force on the suitcase is R. The belt is inclined at an angle of 30° to the horizontal.

Fig. 4.3 shows the suitcase and the forces acting on it at the line labelled YY'.

The centre of mass of the suitcase is now moving at 1.5 m s^–1 along a semi-circular arc of radius 2.0 m.

Calculate the magnitude of the centripetal force acting on the suitcase.

When the suitcase is at line YY', the magnitude of force F is larger and the magnitude of force R is smaller than at XX'.

Explain why this is so.

Describe the Doppler effect.

Explain how ultrasound is used to measure the speed of blood flow in an artery.

Use Fig. 6.2 to show that the speed of the loudspeaker is 7.5 m s^–1. (i)

The speed of sound in this experiment is 330 m s^–1. (ii)

Calculate the maximum change in frequency f of the sound detected by the microphone.

Hence complete the scale on the y-axis of Fig. 6.2.

Mark with an X on Fig. 6.1 the position of the loudspeaker which corresponds to the

point X on Fig. 6.2.

In their laboratory notes, one student writes about the accuracy of the measurements whereas the other writes about their precision.

Define these terms.

A binary star is a pair of stars which move in circular orbits around their common centre of mass.

In this question consider the stars to be point masses situated at their centres.

Fig. 3.1 shows a binary star where the mass of each star is m. The stars move in the same circular orbit.

Explain why the stars of equal mass must always be diametrically opposite as they travel in the circular orbit.

The stars are viewed from Earth in the plane of rotation. The stars are observed using light that has wavelength of 656 nm in the laboratory. The observed light from the stars is Doppler shifted.

Calculate the maximum change in the observed wavelength of this light from the orbiting stars. Give your answer in nm.

The centre of mass of the binary star lies at the surface of the star of mass 4 m. Draw on

Fig. 3.2 two circles to represent the orbits of both stars.

Explain why the smaller mass star travels faster in its orbit than the larger mass star.