Stress, strain & the Young modulus

Which of the following gives the SI base units of stress?

The photograph shows the Petronas Twin Towers in Kuala Lumpur, which are the world’s tallest twin buildings.

Each tower is supported by 16 cylindrical concrete columns.

Calculate the stress at the base of each cylindrical concrete column.

mass of each tower = 3.010⁸kg diameter of each column = 2.4 m

Tall buildings sway in the wind. The design of the Petronas Towers permits safe movement of the towers.

The towers were designed to permit a maximum horizontal displacement of 0.90 m at the top of the towers. The natural frequency of each tower is 0.17 Hz.

Calculate the maximum velocity at the top of one of the towers when the tower is oscillating at its natural frequency.

The twin towers were linked together to improve the response of the building to strong winds.

The graph below shows how the maximum displacement x varies with the frequency f for a single tower and for two linked towers.

Explain the behaviour of a single tower compared with the behaviour of the linked towers.

A wire of length L and diameter d is fixed at one end. A force F is applied to the wire causing an extension e. A second wire, made of the same material, of length 2L and diameter , is fixed in the same way. A force 2 F is applied to this wire.

What is the extension of the second wire?

A student stretched a length of nylon fishing line using the apparatus shown below.

The nylon fishing line was stretched by adding masses to the mass holder. The positions of the bottom of the mass holder were measured as masses were added. For each mass, the extension of the nylon fishing line was calculated. The student recorded the results, as shown in the table.

Criticise the recording of these results.

Describe how the extension of the nylon fishing line could have been determined as accurately as possible.

The strain for the nylon fishing line at its yield point is 0.04

State what is meant by yield point.

The original length of the nylon fishing line was 2.00 m.

Determine whether the fishing line was stretched beyond its yield point.

A wire Y of cross‑sectional area A and length l is joined to a second wire Z of cross‑sectional area 2 A and length 2 l as shown. Wire Z is fixed at one end and a force F is applied to the other end of wire Y.

The wires are made of the same material.

Wire Y extends by a distance x.

Which of the following is the extension of wire Z?

Give one advantage of using a reference wire as well as a test wire.

State why a reference load was applied to the reference wire.

Explain why the test wire should be both long and thin.

The student varied the load F on the test wire and recorded the corresponding change in length Δ x from the vernier scale. The results are shown on the graph.

Δ x / mm

Determine a value for the Young modulus of brass.

length of wire = 2.75 m

The harp is a musical instrument with many strings, as shown.

All the strings are under tension.

The strings on one type of harp are made from nylon of density 1070 kg m⁻³. One string has a diameter of 1.14 mm.

Show that the mass per unit length μ of the string is about 1.1 10⁻³kg m⁻¹.

When the middle of the string is plucked, a note of frequency 440 Hz is produced.

Calculate the tension in the string.

length of string = 41.0 cm

The graph shows how the Young modulus E of the nylon varies with temperature.

When the harp is played, the temperature of the string increases.

Explain how this temperature change would affect the frequency of the note produced when the string is plucked.

A mass of 24 kg is suspended from a steel wire of length 1.5 m. The wire has cross‑sectional area 3.1 10⁻⁶m².

The Young modulus of steel is 1.8 10¹¹Pa.

Which of the following gives the extension of the wire?

The photograph shows a guitar.

When a guitar string is plucked, a standing wave is created.

Explain how a standing wave is created on the string.

The diagram shows a standing wave on a guitar string.

The oscillating length of the guitar string is 66 cm.

State the wavelength for this standing wave.

Calculate the frequency of vibration for this standing wave.

tension in guitar string = 88.6 N

mass per unit length of guitar string = 4.47 10⁻³kg m⁻¹

One end of the guitar string is wrapped around a cylindrical tuning peg. Turning the peg changes the total length of the string and hence changes the tension in the string. This changes the frequency of vibration of the string.

The length of one string is 68 cm.

Calculate the extension required to produce a tension of 93.4 N in the string.

Young modulus of string material = 1.8 × 10⁹N m⁻²

cross-sectional area of string = 6.6 10⁻⁷m²

The vibrating length of string is unchanged by turning the tuning peg.

Explain the effect that tightening the string has on the frequency of the sound produced.

The photograph shows a guitar.

When a guitar string is plucked, a standing wave is created.

Explain how a standing wave is created on the string.

The diagram shows a standing wave on a guitar string.

The oscillating length of the guitar string is 66 cm.

State the wavelength for this standing wave.

Calculate the frequency of vibration for this standing wave.

tension in guitar string = 88.6 N

mass per unit length of guitar string = 4.47 10⁻³kg m⁻¹

One end of the guitar string is wrapped around a cylindrical tuning peg. Turning the peg changes the total length of the string and hence changes the tension in the string. This changes the frequency of vibration of the string.

The length of one string is 68 cm.

Calculate the extension required to produce a tension of 93.4 N in the string.

Young modulus of string material = 1.8 × 10⁹N m⁻²

cross-sectional area of string = 6.6 10⁻⁷m²

The vibrating length of string is unchanged by turning the tuning peg.

Explain the effect that tightening the string has on the frequency of the sound produced.

Show that the length of the wire supporting the sphere is about 10m.

Seat belts are being tested by a car manufacturer. In the test, a car moving at a steady speed of 28 m s^–1 collides with a wall and stops.

A crash-test dummy in the driving seat is wearing a seat belt made from polyester webbing. The seat belt has a cross-sectional area of 0.85 cm² and a total length of 2.0 m. A student suggests that in the collision the seat belt absorbs all the kinetic energy of the dummy.

Show that the energy per unit volume that would have to be absorbed by the seat belt is about 2 10⁸ J m^–3.

mass of dummy = 75 kg

The graph shows how stress varies with strain for the seat belt.

Show that the area under the graph represents the energy stored per unit volume in the seat belt.

Use the graph to determine whether the seat belt absorbs all the kinetic energy of the dummy from part (a).

In this collision, the maximum strain of the seat belt is 0.075

In 2016 scientists at the Laser Interferometer Gravitational‑Wave Observatory (LIGO) announced that gravitational waves had been detected.

The signal they detected is shown on the graph.

Gravitational waves travel at the speed of light.

Determine the mean wavelength of the waves detected between 0.30 s and 0.35 s on the graph.

Gravitational waves alternately compress and stretch matter by very small amounts as they pass through.

The LIGO detector has two arms, at 90° to each other, each 4 km long. As a gravitational wave passes the detector, the arms change length. The detector continuously compares the lengths of the two arms.

An article states that ‘the maximum change in the 4 km length of the arm is about 0.001 times the diameter of a proton’.

Determine whether this statement applies to the gravitational wave shown in the graph.

diameter of proton = 8.810^–16m

In the LIGO detector, any change in the lengths of the arms is detected using a laser beam and photodetector.

The laser beam is split into two at B, one beam travelling to one mirror and the other beam travelling to the other mirror. After reflection at the mirrors, the beams are recombined at B and reach the photodetector. The photodetector measures the intensity of the incident light.

The system is arranged so that when no gravitational waves are present, the beams have a path difference of half a wavelength at the photodetector.

Explain how the photodetector detects very small changes in the length of one arm, when the other arm stays the same length.

The system could be arranged so that when no gravitational waves are present, the beams have zero path difference at the photodetector.

Explain whether using an initial path difference of half a wavelength is a more sensitive way of detecting changes in length than having an initial path difference of zero.