Elastic & Plastic Behaviour (CIE AS Physics)

An adventure experience is shown in Fig. 1.1 below.

Fig. 1.1

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 27 m. When stretched, the rope obeys Hooke’s law. The rider and harness have a total mass of 65 kg. 

The rider is initially held at rest at ground level. The lower end of 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, is adjusted so the rope becomes unstretched when the rider is at the highest point of the ride. The rider is released and moves upwards, reaching a maximum height when the rope is at its unstretched (natural) length. The rider then oscillates vertically. 

The height of point A above the ground is 51 m. 

Calculate the spring constant of the rope neglecting air resistance and ignoring the mass of the rope in this question.

A different ride consists of a 15 m length of rope, stretched into a ‘V’ shape PQR on a frame, as shown in Fig 1.2. The frame is 16 m wide and the rider stands in the middle at position R. The ends of the elastic rope are fixed to the frame at the point P and Q. 

Fig. 1.2

The rider is attached to the midpoint of the elastic rope at R. The Young Modulus of the rope is 1920 N m⁻¹.

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

The elastic bungee cord is replaced with an elastic which can also provide mechanical resistance when performing fitness exercises. The graph of load against extension for the chord is shown in Fig. 1.3 below. 

Fig. 1.3

Estimate the minimum energy required to stretch the elastic to an extension of 0.40 m.

Fitness equipment often uses springs as a method of creating resistance. One example is the Push Down Bar, which the manufacturer says can help a person who uses it to 'increase strength and burn calories'.

To use a Push Down Bar a person applies force by pushing the handles towards each other as shown in Fig. 1.1.

Fig. 1.2 shows the construction of the Push Down Bar.

The relationship between applied force and change in length of the springs was measured for a range of values of force.

The results are plotted on the graph in Fig. 1.3.

Derive the formula for the elastic potential energy stored by the spring from the graph of force against extension.

State and explain whether the material chosen for the spring for the equipment in Fig. 1.1 should exhibit elastic or plastic behaviour.

A manufacturer produces springs for use in school laboratory investigations. As part of quality control the springs are spot-checked  by measuring the extension produced for certain applied loads.

The graph of the testing data is shown in Fig. 1.1

Fig. 1.1

Use a graphical method to calculate the spring constant of the spring.

Show your working.

Show that the work done in extending the spring up to point A is approximately 0.5 J.

When the spring reaches an extension of 0.046 m, the load on it is gradually reduced to zero.

On the graph in Figure 1.1, sketch how the extension of the spring will vary with load as the load is reduced to zero.

Explain why the graph has this shape.

Without further calculation, compare the total work done by the spring when the load is removed with the work that was done by the load in producing the extension of 0.046 m.  

Explain how this is represented on the graph drawn in part (c).

Materials scientists are asked to produce a material which would be suitable to use in constructing the wing of an aeroplane.

The team of scientists work with two materials, an aluminium alloy and a carbon fibre composite.

Suggest a material property which the scientists are likely to investigate and report on, and give a reason.

Fig. 1.1 shows the stress-strain graph that the scientists obtain for the aluminium alloy.

They have labelled a point Q on the graph.

The aluminium alloy wire has a diameter of 1.2 mm. Calculate the elastic potential energy stored in the wire when it shows an extension of 7.5 cm under a total strain of 190 MPa. 

A steel bar is 40 mm long and has cross-sectional area 4.5 × 10⁻⁴ m².

The bar is compressed using a vice so that the length is reduced by 0.20 mm.

The Young Modulus of steel = 2.1 × 10¹¹ Pa.

Calculate the compressive force exerted on the bar.

Calculate the work done compressing the bar.

For the compression described in part (a)