Resultant forces

A light rod is hung from a horizontal ceiling using 2 springs, one attached to each end. The springs have force constants k and 2 k.

A downwards force F is applied closer to one end of the rod such that the rod is horizontal.

What is the tension T in the spring with the force constant 2 k ?

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

A wooden block is stationary on a ramp.

The diagram is not drawn to scale.

The block has weight W. The normal contact force on the block is N. The frictional force F on the block is not shown on the diagram.

Which triangle of forces diagram is correct?

Your answer

A truck pulls a car up a slope at a constant speed. The truck and the car are joined with a steel tow bar, as shown in the diagram.

The diagram is not drawn to scale.

The slope is 10 to the horizontal ground. The mass of the car is 1100 kg. The car travels from A to B. The vertical distance between A and B is 120 m.

There are four forces acting on the car travelling up the slope.

Complete the free-body diagram below for the car and label the missing forces.

Show that the component of the weight of the car W_s acting down the slope is about 1900 N.

The total frictional force acting on the car as it travels up the slope is 300 N.

Calculate the force provided by the tow bar on the car.

The steel tow bar used to pull the car has length 0.50 m and diameter 1.2 × 10^–2 m. (e) The Young modulus of steel is 2.0 × 10¹¹ Pa.

Calculate the extension x of the tow bar as the car travels up the slope.

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.

An object is falling. The weight of the object is 4.5 N. The wind provides a horizontal force of magnitude F on the object. The resultant force on the object is 5.8 N. Air resistance and upthrust on the object are negligible.

What is the value of F ?

A 1.3 N

B 3.7 N

C 7.3 N

D 13 N

Your answer

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.

A swimming pool designer investigates the depth d below a water surface reached by a diver when diving from a height h above the water surface. The designer models the diver as a uniform wooden cylinder. The experimental arrangement is shown in Fig. 18.1.

Calculate the density of the wood.

density =........ kg m⁻³

Suggest why wood is an appropriate material to model the depth reached by a diver.

The cylinder is released from rest from a trapdoor. The base of the cylinder is at a height h = 0.30 m above the water surface. Calculate the speed of the cylinder just before the base hits the water. Ignore air resistance.

speed =........ m s⁻¹

Fig. 18.2 shows the cylinder fully submerged under the water surface before it has come to rest. The cylinder is moving vertically down.

Add arrows to Fig. 18.2 to show the three forces acting on the wooden cylinder. Label

the arrows.

Describe and explain how the resultant force on the wooden cylinder varies from the moment the cylinder is fully submerged until it reaches its deepest point.

The graph of Fig. 18.3 shows the depth d reached for different initial drop height h.

The designer is required to double the height of a diving board for an existing swimming pool. He suggests that the depth of the pool also needs to be doubled. Use Fig. 18.3 to explain whether you agree with this suggestion.

Fig. 20.1 shows a positively charged metal sphere and a negatively charged metal plate.

On Fig. 20.1, draw a minimum of five electric field lines to show the field pattern between the

sphere and the plate.

Define electric potential at a point in space.

A metal sphere is given a positive charge by connecting its surface briefly to the positive terminal of a power supply. The electric potential at the surface of the sphere is + 5.0 kV. The sphere has radius 1.5 cm.