4.7 Bulk Properties of Solids

A light spring of length 1.5 cm is hung from a fixed point. An object of weight 4.0 N is hung from the other end of the spring. Figure 1a shows the length of the spring when the object is in equilibrium. 

Figure 1a

The object is then pulled vertically downwards and is no longer in its equilibrium position. This is shown in Figure 1b. 

Figure 1b

The change in elastic strain energy ΔE between the spring in Figure 1a and Figure 1b is 0.25 J. 

   Calculate the length of the spring in Figure 1b.

The object is released from its position shown in Figure 1b. 

   Calculate its initial acceleration.

Figure 2 shows a force–extension graph for the spring. 

Figure 2

Assuming that the spring is not extended beyond its limit of proportionality

Figure 3 now shows two different springs and  . The top ends of the springs are attached to a rod. 

Figure 3

A mass is hung from the bottom end of . The extension of  is x and the elastic potential energy in the spring is 25 mJ. The same mass is hung from the bottom end of . The extension of  is  and its spring constant is 3 times that of . 

   Calculate the elastic potential energy in .

The ultimate tensile stress for a hollow steel column which carries an axial load of 300 MN is 0.50 GPa. The external diameter of the column is 667 cm. 

Calculate the internal diameter. State your answer to an appropriate number of significant figures.

In a reverse bungee experience a ‘rider’ is catapulted high into the air.

A designer creates a less extreme version for more timid participants, as shown in Figure 1 below. 

Figure 1

The rider is strapped into a rigid harness attached to one end of an elastic rope AB. The rider and the rope behave in the same way as a mass–spring system. 

The rope has an unstretched length of 23 m. When stretched, the rope obeys Hooke’s law. The rider and harness have a total mass of 70 kg. 

The rider is initially held at rest at ground level. The lower end of the rope B is attached to the rigid harness at a point which is 3.1 m above the ground, so the harness doesn’t injure the rider by being too close. The top end of the rope, A, has to be adjusted so that the rope just becomes unstretched when the rider is at the highest point of the ride. The rider is then released and moves upwards, reaching a maximum height when the rope is at its unstretched (natural) length. The rider then oscillates vertically until eventually coming to rest, suspended above the ground. 

The height of point A above the ground is 42 m. 

   Calculate the spring constant of the rope. 

   Neglect air resistance and ignore the mass of the rope in this question.

A different reverse bungee for braver participants consists of a 15 m length of rope, that is stretched into a ‘V’ shape PQR on a frame, as shown in Figure 2. The ends of the elastic rope are fixed to the frame at the point P and Q. 

Figure 2

The rider is attached to the midpoint of the elastic rope at R. The Young Modulus of the rope is 1750 N. 

Calculate the elastic potential energy of the elastic rope in the initial position shown in the diagram.

The mass of the rider and their harness is 53 kg and can be treated like a point mass. The rider is in equilibrium with the tension in the elastic rope. The rope has an ultimate tensile stress (UTS) of 15 MPa. 

Calculate the minimum diameter of the rope that could be used to keep the rider in this position.

The elastic bungee cord is replaced with a rubber which can also be used to provide mechanical resistance when performing fitness exercises. A student decided to test the properties of the cord to find out how effective it was for this purpose. The graph of load against extension is shown in Figure 3 below for a 0.50 m length of the cord. 

Figure 3

Estimate the work done in order to permanently deform the rubber cord.

A steel ball of density 8.05 g cm^–3 has a diameter of 6.50 cm. 

Calculate the weight of this ball.

A seismometer is a device that is used to record the movement of the ground during an earthquake. A simple seismometer is shown in Figure 1. 

Figure 1

The steel ball from part (a) is attached to a pivot by a rod so that the rod and ball can move in a vertical plane. The rod is suspended by a spring so that, in equilibrium, the spring is vertical and the rod is horizontal. A pen is attached to the ball. The pen draws a line on graph paper attached to a drum rotating about a vertical axis. Bolts secure the seismometer to the ground so that the frame of the seismometer moves during the earthquake. 

When the rod is horizontal and the spring is vertical, the spring extends by 8.3 cm. 

   Show that the spring constant of the spring is around 240 N m^–1.

Discuss why it is important that the spring constant of the spring is not too low or high.

Figure 1 shows the variation of tensile stress with tensile strain for two wires X and Y, which have the same dimensions, but are made of different materials. The materials fracture at the points F_x and F_Y respectively. 

Figure 1

State, with a reason, two properties of wire X.

State, with a reason, two properties of wire Y.

Wire X originally has a length of 1.35 m. A force of 200 N is used to extend wire X to 1.51 m. 

Calculate the energy stored in wire X if it extends a further 30 mm.

State and explain which wire would be more suitable for use as cables and structural beams. 

A type of exercise device is used to provide resistive forces when a person applies compressive forces to its handles. The stiff spring inside the device compresses as shown in Figure 1.

Figure 1

The force exerted by the spring over a range of compressions was measured. The results are plotted on the graph shown in Figure 2. 

Figure 2

State, with a reason, whether the spring obeys Hooke’s law over the range of values tested.

Use the graph in Figure 2 to calculate the spring constant, stating an appropriate unit.

Derive the formula for the energy stored by the spring using a graph of force against extension.

State and explain whether the material chosen for the spring for the device in Figure 1 should exhibit elastic or plastic behaviour.

Nichrome is a common alloy used to make coins and heating elements in electrical appliances. It consists of 80 % by volume of nickel and 20 % by volume of chromium. 

Determine the mass of nickel and the mass of chromium required to make a wire of nichrome of volume 0.50 × 10^-3 m³.           

Density of nickel = 8.9 × 10³ kg m^–3

Density of chromium = 7.1 × 10³ kg m^–3

Calculate the density of nichrome.

The radius of the nichrome wire is 5.7 mm. The wire breaks when a force of 72 kN is applied to the wire. 

Calculate the breaking stress of nichrome, stating an appropriate unit.

Hence, or otherwise, explain why nichrome is a suitable material for making coins.

Figure 1 shows a bungee jumper about to step off a raised platform. The jumper comes to a halt for the first time when their centre of mass has fallen through a distance of 25 m. The bungee rope has an unextended length of 18 m and a stiffness of  410 N m^–1. 

Ignore the effects of air resistance and the mass of the rope in this question. Treat the jumper as a point mass located at the centre of mass. 

 Figure 1

Using the extension of the bungee rope, calculate the resultant force acting on the jumper when they reach the lowest point in the jump.

The extension of the rope is 1.6 m when the acceleration of the jumper is zero. 

Calculate the mass of the bungee jumper.

The extension of the bungee rope is 2.4 m when the jumper’s centre of mass has fallen through a distance of 16 m. 

Use the principle of conservation of energy to calculate the speed of the jumper in this position.         

The bungee jump operator intends to use a bungee rope of the same unextended length but with much less stiffness. The rope is to be attached in the same way as before. 

Explain, with reference to the kinetic energy of the jumper, how the force on the jumper will be affected by this change as they are slowed down by the new rope.