Stress, strain and the Young modulus

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.

Which row in the table correctly identifies the elastic limit, fracture and ultimate tensile strength in

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 graph showing the variation of the stress σ with strain ε for a material is shown below.

What is the Young modulus of the material?

6.0 × 10⁴ Pa A

1.2 × 10⁹ Pa B

8.0 × 10¹⁰ Pa C

1.2 × 10¹¹ Pa D

Your answer

Explain what is meant by the ultimate tensile strength of a material.

A footbridge is supported by a number of metal cables of the same length. Each cable has uniform cross-section and diameter 4.20 mm as shown in Fig. 16.1.

A group of engineers investigate how the extension x varies with applied force F for one of the cables.

The results of the investigation are shown in Fig. 16.2.

The cable breaks when the force is 2.2 kN.

Describe how a suitable measuring device may have been used by the engineers to demonstrate that the cable had uniform cross-section.

State any value of F when the cable behaves

1. elastically

2. plastically.

Determine the breaking stress of the cable. Assume that the cross-sectional area of the cable remains constant during the test.

Explain why the work done on the cable when its extension changes from 3.0 mm to 4.0 mm is greater than when its extension changes from 1.0 mm to 2.0 mm.

Four materials A, B, C and D have the same length and cross-sectional area. The force against extension graph for each material up to the breaking point is shown below.

Which material is brittle and has the greatest ultimate tensile strength?

Your answer