Moments, levers and gears

Part of a machine to generate electrical energy using waves is a cylindrical drum whose cross-section is shown below. The height of the drum is 2.0 m and its cross-sectional area is 0.79 m².

The mass of the drum is 1200 kg.

The centre of mass of the drum is 0.50 m from one of the circular faces, on a line joining the centres of the circular faces.

The density of water is 1000 kg m^–3.

The drum floats in flat water with its long axis vertical. The base of the drum is a distance d below the surface, where d is measured in m.

Explain why the weight of water displaced must be equal to the weight of the drum.

Calculate d.

Upthrust forces act at the centre of mass of the fluid displaced.

Explain why the drum cannot float in equilibrium on its side with the long axis horizontal.

When the vertical drum is displaced a distance x deeper into the water from the equilibrium position, the magnitude of the resultant force on the drum, F, is given by

F = –kx

where k is a constant and d is measured in m.

By considering the magnitude of the additional upthrust on the drum, show that k is about 8000 Nm^–1.

When the drum is released, it oscillates vertically. Explain why the oscillations are simple harmonic motion.

Calculate the natural frequency f of the oscillations.

* Two designs, A and B, of the machine are tested in a large ripple tank.

The ripple tank can produce waves whose amplitude and frequency can be varied over wide ranges.

The natural frequencies of the two machines are identical. The graphs show the results of the tests but are incomplete.

• Describe how the amplitude of the machine could be measured at each frequency tested. You do not need to describe how to vary the amplitude or frequency of the waves.

• Complete the graphs with suitable labels and annotations.

• Discuss which of the designs is the most effective at generating electrical energy.

Additional space if required

An electroscope is an instrument used to measure electric charge.

The diagram shows one type of electroscope.

The diagram below shows the needle of the electroscope when it is negatively charged. The centre of mass, X, of the needle has been marked on the diagram.

Suggest why the centre of mass of the needle is a short distance below the pivot.

A student charges the electroscope by connecting the metal plate to the negative terminal of a high voltage power supply.

They record the deflection angle of the needle for a range of values of the voltage measured using a voltmeter.

Their results are shown in the graph below.

The student concludes that the voltmeter has a zero error of 300 V.

Explain their reasoning.

The capacitance of the electroscope is 20 pF.

Estimate the number of electrons stored on the electroscope when the deflection angle is 45°.

Describe how to find the centre of mass of a 2-dimensional shape, including any equipment required.

Fig. 19 shows a bridge.

The bridge can be raised by an electric motor to allow tall ships to pass underneath.

The moving section of the bridge is 14 m long and has a weight of 120 kN.

The centre of mass of the structure is 6.3 m from the pivot.

Calculate the average power required to raise the bridge to a vertical position in 90 s.

Suggest why the actual electric motor used to lift the bridge has a maximum power output several times larger than the value calculated in (b)(i).

A ball coated with conducting paint has weight 0.030 N and radius 1.0 cm. The ball is suspended from an insulating thread. The distance between the pivot and the centre of the ball is 120 cm.

The ball is placed between two vertical metal plates. The separation between the plates is 8.0 cm. The plates are connected to a 4.0 kV power supply.

Show that the electric force acting on the charged ball is 4.5 × 10^–4 N.

Draw, on the diagram above, arrows which represent the three forces acting on the ball.

Label each arrow with the name of the force it represents.

By taking moments about the pivot, or otherwise, show that x = 1.8 cm.

A screw is used to hang a wooden sign on a wall. It is screwed into the wall using a screwdriver.

The width of the screwdriver blade is 5.0 × 10^–3 m from end to end. (a) The ends of the blade exert equal and opposite forces on the screw. The magnitude of each force is 350 N, as shown in Fig. 22.1.

Calculate the magnitude of the torque of the couple produced by the forces at each end of the screwdriver blade.

torque =........ N m

The wooden sign is then hung on the screw at point A. The forces acting on the screw are shown in Fig. 22.2.

The inside section of the wall exerts a maximum downwards force of 31 N at a distance of 3.0 × 10^–2 m from the outer edge of the wall. The hanging wooden sign exerts a force F at a distance 7.0 × 10^–3 m from the outer edge of the wall. There is a force R acting on the screw at the outer edge of the wall. The mass of the screw is negligible.

Use the principle of moments to calculate the maximum mass of the wooden sign.

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.