Stress, strain and the Young modulus

Explain why the Young modulus and the breaking stress of a wire have the same SI unit.

Table 1 contains data about four metal wires W, X, Y and Z.

The metal used to make wire X costs £75 per gram.

Calculate, in £, the cost of wire X.

State and explain which of the wires in Table 1 are made from the same material.

One of the wires in Table 1 was used to obtain the stress–strain data shown in Figure 5.

Draw a line of best fit on Figure 5. Go on to deduce, using your line of best fit and Table 1, whether wire W could have been used to produce Figure 5.

Figure 11 shows the stress–strain graph for a metal in tension up to the point at which it fractures.

Determine, using Figure 11, the Young modulus of the metal.

Explain how the graph shows that this metal is brittle.

Figure 12 shows a uniform rigid lighting beam AB suspended from a fixed horizontal support by two identical vertical steel wires. A lamp is attached to the midpoint of AB

The unloaded length of each steel wire was 1.20 m before it was attached to AB. AB is horizontal.

mass of AB = 4.4 kg mass of lamp = 16.0 kg distance between wires = 2.00 m diameter of each wire = 0.800 mm Young modulus of steel = 2.10 × 10¹¹ Pa

Calculate the extension of each wire.

The right-hand steel wire is removed and replaced with an aluminium wire of diameter 1.60 mm. The unloaded length of the aluminium wire is the same as that of the original steel wire.

When the lamp is at the midpoint of AB, one of the wires extends more than the other so that AB is not horizontal. To make AB horizontal the lamp has to be moved to a distance x from A. Figure 13 shows the new arrangement.

The Young modulus of aluminium is 7.00 × 10¹⁰ Pa.

Deduce distance x.

Two wires P and Q are made of the same material and have the same cross-sectional area. P has an original length L and is subject to a tensile force F. P extends a distance x. Q has an original length 2 L and is subject to a tensile force 2 F.

Which statement is correct?

Figure 5 shows a strip of steel of rectangular cross-section clamped at one end. The strip extends horizontally over the edge of a bench.

A mass m is suspended from the free end of the strip. This produces a vertical displacement y. A student intends to measure y with the aid of a horizontal pin fixed to the free end of the steel strip. She positions a clamped vertical ruler behind the pin, as shown in Figure 6.

Explain a procedure to avoid parallax error when judging the reading indicated by the position of the pin on the ruler. You may add detail to Figure 6 to illustrate your answer.

It can be shown that

where:

L is the distance between the free end of the unloaded strip and the blocks w is the width of the strip and is approximately 1 cm t is the thickness of the strip and is approximately 1 mm E is the Young modulus of the steel.

A student is asked to determine E using the arrangement shown in Figure 5 with the following restrictions:

• only one steel strip of approximate length 30 cm is available

• m must be made using a 50 g mass hanger and up to four additional 50 g slotted masses

• the experimental procedure must involve only one independent variable

• a graphical method must be used to get the result for E.

Explain what the student must do to determine E.

Two wires X and Y have the same extension for the same load. X has a diameter d and is made of a metal of density ρ and Young modulus E. Y has the same mass and length as X but its diameter is 2 d.

What are the density and the Young modulus of the metal from which Y is made?

An experiment is carried out to determine the Young modulus E of steel using a vertical wire of initial length L and cross-sectional area A. Various weights are suspended from the wire. A graph of extension against weight is plotted.

What does the gradient of the graph represent?

Figure 5 shows apparatus used to investigate the bending of a beam.

The beam is placed horizontally on rigid supports. The distance L between the supports is 80 cm.

A travelling microscope is positioned above the midpoint of the beam and focused on the upper surface.

Figure 6 shows an enlarged view of both parts of the vernier scale.

The smallest division on the fixed part of the scale is 1 mm.

What is the value of the vernier reading R₀ in mm? Tick (✓) one box.

Figure 7 shows the beam bending when a hanger of mass 0.050 kg is suspended from the midpoint.

The microscope is refocused on the upper surface and the new vernier reading R is recorded. The vertical deflection s of the beam is equal to (R – R₀).

The total mass m suspended from the beam is increased in steps of 0.050 kg. A value of s is recorded for each m up to a value of m = 0.450 kg. Further values of s are then recorded as m is decreased in 0.050 kg steps until m is zero.

Student A performs the experiment and observes that values of s during unloading are sometimes different from the corresponding values for loading.

State the type of error that causes the differences student A observes.

Student B performs the experiment using a thinner beam but with the same width and made from the same material as before.

Discuss one possible advantage and one possible disadvantage of using the thinner beam.

Figure 8 shows the best-fit line produced using the data collected by student A.

where E is the Young modulus of the material of the beam and η is a constant.

Deduce in s⁻² the order of magnitude of η.

E = 1.14 GPa

Student C performs a different experiment using the same apparatus shown in Figure 5 on page 10. A mass M is suspended from the midpoint of the beam. The vertical deflection s of the beam is measured for different values of L.

Figure 9 shows a graph of the results for this experiment.

Figure 9 shows that log₁₀(s / m) varies linearly with log₁₀(L / m).

State what this shows about the mathematical relationship between s and L. You do not need to do a calculation.

Deduce, using Figure 9, the value of s when L = 80 cm.

Determine M using Figure 8.

Which combination of properties would produce the smallest extension of a wire when the same tensile force is applied to the wire?

A steel wire W has a length l and a circular cross-section of radius r. When W hangs vertically and a load is attached to the bottom end, it extends by e. Another wire X made from the same material has the same load attached to it. Which length and radius for X will produce an extension ?

What is the Young modulus of the second sample of wire in terms of E ?