4.8 The Young Modulus

Table 1 show the results of an experiment where a force was applied to a copper wire. 

Table 1

On the axes shown in Figure 1, plot the graph of stress against strain using the data in the table for the copper wire. 

Figure 1

Use the graph drawn in part (a) to calculate the Young Modulus of copper. State an appropriate unit for your answer.

Use the graph in drawn in part (a) to find the work done per unit volume in stretching the copper wire to a strain of 3.5 × 10^–3.

A certain material has a Young Modulus greater than copper and undergoes brittle facture at a stress of 182 MPa. 

On the graph drawn in part (a), draw a line showing the possible variation of stress with strain for this material.

As part of a quality check, a manufacturer of fishing rods selects a sample of the lines used for the rods to a tensile test. 

The sample of line used is 2.0 m long and is of constant circular cross–section of diameter 0.55 mm. Hooke’s law is obeyed up to the point when the line has been extended by 50mm at a tensile stress of 1.9× 10⁸ Pa.

The maximum load the line can support before breaking is 55 N at an extension of 9 cm. 

Sketch a graph on Figure 1to show how you expect the tensile stress to vary with strain. Mark the value of stress and corresponding strain at 

(i)      The limit of Hooke’s law, 

(ii)     The breaking point. 

Figure 1

A different 2.0 m fishing line sample has a larger diameter. It is made of the same material and is subject to the same force. 

Compare the advantages and disadvantages of this fishing line compared to the first.

Calculate the force needed to extend a piece of 3 m nylon fishing wire with a radius of 1.2 mm to change its surface area by 3.5 × 10^–3 m². Assume that the cross–sectional area is kept constant throughout the extension. 

            Young Modulus of nylon = 2.7 GPa.

Figure 1 shows two wires, one made of Molybdenum and the other of Titanium, firmly clamped together at their ends. The wires have the same unstretched length and the same cross–sectional area.

One of the clamped ends is fixed to a horizontal support and a mass M is suspended from the other end, so that the wires hang vertically. 

Figure 1

The mass M is 2 kg. 

Calculate the force on the titanium wire, . 

            Young Modulus of Molybdenum,  = 330 GPa

            Young Modulus of Titanium,  = 110 GPa

Two spring constants and combine in parallel in the following way:   where  is the equivalent spring constant of the combination. 

Calculate the equivalent Young Modulus of the combination of the Molybdenum and Titanium wires. Assume this time that the force on each wire is identical.

Figure 2 now shows the Molybdenum and Titanium wires joined together, now at different lengths but still with the same cross–sectional area. This combination is suspended from a fixed support and a force F is applied at the bottom end. 

Figure 2

The two wires extend by a total of 5.4 × 10^–4 m. 

Length of the Molybdenum wire = 0.90 m

Length of the Titanium wire = 1.25 m 

Calculate the individual extensions of the Molybdenum and Titanium wires.

The total mass of the combination of wire is 5.3 g. 

Show that the radius of both wires is around 0.34 mm. 

            Density of Molybdenum = 10 000 kg m^–3

            Density of Titanium = 4420 kg m^–3

Calculate the new mass of the wire if the diameter of both wires is twice as large.

In order to prevent the collapse of walls of old buildings a metal rod is often used to tie opposite walls together, as shown in Figure 1 below. 

Figure 1

In one case a steel tie rod of diameter 24 mm is used. When the nuts are tightened, the rod extends by 7 %. The Young Modulus of steel is 2.1 × 10¹¹ Pa. 

Calculate the force exerted on the walls by the rod.

Another part of the old building requires 6 steel bolts, each of diameter 8.5 cm. When an earthquake occurs, this produces a strain on each bolt of 5.3 × 10^–4. 

Calculate the maximum force exerted on the bolts during this earthquake.

The ultimate tensile stress of steel is 5.0 × 10⁸ Pa. 

Determine the minimum number of bolts that is required to make sure the building stays intact during this earthquake.

Figure 1 shows a tower crane that has three identical steel cables. The length of each steel cable is 41 m from the jib to the hook. 

Figure 1

The cables would break if the crane attempted to lift a load of 2.7 MN or more. 

Calculate the breaking stress of one 

Radius of each cable = 12 mm.  

When the crane supports a load each cable experiences a stress of 500 MPa. Each cable obeys Hooke’s law. 

Calculate the mass of the load.

The unstretched length of each cable is 41 m. 

Calculate the final length of each cable when supporting the load. Give your answer to three significant figures. 

Young Modulus of steel = 2.1 × 10¹¹ Pa

Calculate the combined stiffness constant, k, for the three cables. State an appropriate unit in your answer.      

One end of a steel wire of length 1.5 m and 3.0 mm diameter is attached to a rigid beam. A 32 g mass is attached to the free end of the steel wire and placed against the underside of the beam as shown in Figure 1. 

Figure 1

The mass is released and falls freely until the wire becomes taut. The gravitational potential energy of the falling mass is converted to elastic potential energy in the wire as the wire extends to a maximum of 2.0 mm. Energy converted to other forms is negligible. 

Calculate the tension in the wire.

Calculate the stress in the wire.

Calculate the strain in the wire.

Hence, or otherwise, calculate the Young Modulus in the wire.

If lengths of rail track are laid down in cold weather, they may deform as they expand when the weather becomes warmer. Therefore, when rails are laid in cold weather they are stretched and fixed into place while still stretched. This is called pre-straining. 

The following data is typical for a length of steel rail: 

Young modulus of steel = 2.0 × 10¹¹ Pa
Amount of pre-strain =  3.5 × 10^–5 for each kelvin rise in temperature the rail is expected to experience. 

A steel rail is laid when the temperature is 6 °C and the engineer decides to use a pre-strain of 1.24 × 10^–4. The tensile force required to make this pre-strain is 0.19 MN. 

Show that the cross-sectional area of a length of rail is about 7.7 × 10^-3 m².

The elastic strain energy stored in the rail when pre-strained as in part (a) is 4.2 kJ. 

Calculate the original unstressed length of the rail.

Calculate the temperature at which the steel rail becomes unstressed.

State and explain whether the pre-strain that the engineer decided to use is suitable in all-weather conditions.

A rope climber of mass 70 kg has a rope of length 2.3 m and 1 cm diameter attached to their harness. The other end of the rope is attached to a ridge on a rock face and the climber can be treated like a point mass. 

The climber falls freely vertically downwards until the rope is taut as shown in Figure 1. The gravitational potential energy of the falling climber is converted to elastic potential energy in the rope as the rope extends to a maximum of 1.5 mm. Energy converted to other forms is negligible. 

Figure 1

Complete parts (a) to (d) using the maximum extension of the rope 

Calculate the tension in the rope.

Calculate the stress in the rope.

Calculate the strain of the rope.

Hence, calculate the Young modulus of the material of the rope.