Forces and elasticity

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.

A uniform wire is stretched by a load F. The elastic strain energy stored in the wire is E.

The load is increased from F to 2 F. The wire obeys Hooke’s law.

What is the increase in the elastic strain energy stored in the wire?

A spring is compressed by a force F. The spring has stiffness k and its length changes by Δ L during the compression. When the force is removed the spring returns to its original length in time t.

What is the average power developed by the spring as it returns to its original length?

When a mass M is suspended from a spring, the spring extends by a distance x. M is displaced vertically and, when released, it oscillates with a period T. M is removed and suspended from a different spring. The spring extends by a distance M is again displaced vertically and released.

What is the new period of oscillations of M ?

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.

A heavy cable is attached to a fixed support and carries a load at its lower end.

The weight of the cable is not negligible. The cable has constant cross-sectional area and density.

Which graph shows the variation of tensile stress σ in the cable with distance d from J to K ?

Figure 3 shows a garden gate with a pulley system designed to close the gate.

The pulley system raises weight A when the gate is opened. When the gate is released, A falls. The horizontal cable C passes over pulley R. The tension in cable C causes the gate to close.

Weight A is a solid cylinder with the following properties:

length = 0.23 m weight = 35 N

Table 2 gives the density of three available materials.

Deduce which one of the three materials is used for A.

Figure 4 shows the pulley arrangement when the gate is closed.

Pulleys P and M are frictionless so that the tension in the rope attached to A is equal to the weight of A. A weighs 35 N and the weight of moveable pulley M is negligible.

Calculate the tension in the horizontal cable C when the gate is closed.

Pulley M is pulled to the left as the gate is opened.

Explain why this increases the tension in the horizontal cable C.

Figure 5 shows a plan view with the gate open. The horizontal cable C passes over pulley R and is attached to the door at D. The angle between the door and the horizontal cable C is 12°.

The horizontal distance between the hinge and D is 0.95 m.

The tension in the horizontal cable C is now 41 N.

Calculate the moment of the tension about the hinge.

The same system is attached to an identical gate with stiffer hinges. Now the system does not supply a sufficiently large moment to close the gate.

Discuss two independent changes to the design to increase the moment about the hinges due to horizontal cable C.

A mass M is suspended from a spring. When the mass is at rest at the equilibrium position, the elastic potential energy stored is E. An extra mass of 2 M is added to the spring and the spring extends while still obeying Hooke’s law.

What is the total elastic energy stored when the system is at rest at the new equilibrium

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 6 shows a rotating spacecraft that is proposed to carry astronauts to Mars.

The spacecraft consists of two parts A and B connected by a rigid cylindrical rod. When the spacecraft is travelling, A and B rotate at a constant angular speed about their common centre of mass O.

L is the distance between the centre of mass of A and the centre of mass of B. r_A is the distance from O to the centre of mass of A.

As the spacecraft rotates, a force that imitates the effect of gravity acts on an astronaut who is in contact with the floor.

Explain why.

The forces exerted on A and B by the connecting rod have the same magnitude.

Show, by considering the centripetal forces acting on A and B, that r_A is given by

In this spacecraft m_A < m_B.

Deduce whether the centre of mass of A or the centre of mass of B rotates with a greater linear speed.

The astronauts live in A and the cargo is stored in B.

When loaded,

The spacecraft imitates the gravity of Mars where g = 3.7 m s⁻².

Figure 7 shows a stress–strain curve for the metal used for the rigid rod.

Suggest a suitable diameter for the rod. Justify your answer.

A 0.20 kg mass is suspended from a spring. A 0.10 kg mass is suspended from the 0.20 kg mass using a thread of negligible mass. The system is in equilibrium and the thread is then cut.

What is the upward acceleration of the 0.20 kg mass at the instant that the thread is cut?

A load of 50 N is suspended from a wire that has an area of cross-section of 1 mm².

The stress in the wire, in Pa, is between

A stationary wave is formed on a stretched wire.

Figure 14 shows the wire, fixed at one end, supported by two bridges and passing over a pulley.

A 0.500 kg mass is attached to the free end of the wire. A uniform horizontal magnetic field is applied perpendicular to the wire between the bridges. A signal generator is connected to each end of the wire.

The oscilloscope shown is used to determine the frequency of the output of the signal generator. The wire oscillates because the alternating current in the wire interacts with the magnetic field.

Figure 15 shows the first harmonic stationary wave produced when the distance x between the bridges is adjusted.

The output potential difference (pd) of the signal generator is displayed on the oscilloscope, as shown in Figure 16.

The time-base setting of the oscilloscope is 10 ms cm¹.

Determine f, the frequency of the alternating pd.

A metre ruler is placed next to the bridges supporting the wire, as shown in Figure 17

Determine c.

Determine, in kg m¹, the mass per unit length of the wire.

A student uses digital vernier callipers to measure the diameter of a cylindrical metal rod. The student places the rod between the jaws of the callipers and records the reading indicated. Without pressing the zero button, the student removes the rod and closes the jaws.

Figure 18 shows the calliper readings in millimetres, before and after the jaws are closed.

Calculate the diameter d of the rod.

Describe relevant procedures to limit the effect of random error in the result for d.

Determine the density of the rod. The mass per unit length of the rod is 3.54 10³ kg m¹.

Figure 4 shows the structure of a violin and Figure 5 shows a close-up image of the tuning pegs.

The strings are fixed at end A. The strings pass over a bridge and the other ends of the strings are wound around tuning pegs that have a circular cross-section. The tension in the strings can be increased or decreased by rotating the tuning pegs.

Explain how a stationary wave is produced when a stretched string is plucked.

The vibrating length of one of the strings of a violin is 0.33 m When the tension in the string is 25 N, the string vibrates with a first-harmonic frequency of 370 Hz

Show that the mass of a 1.0 m length of the string is about 4 × 10^–4 kg

Determine the speed at which waves travel along the string in question 03.2 when it vibrates with a first-harmonic frequency of 370 Hz

Figure 6 shows how the tension in the string in question 03.2 varies with the extension of the string.

The string with its initial tension of 25 N is vibrating at a frequency of 370 Hz The diameter of the circular peg is 7.02 mm

Determine the higher frequency that is produced when the string is stretched by rotating the tuning peg through an angle of 75°

Assume that there is no change in the diameter of the string.

A non-uniform sign is 0.80 m long and has a weight of 18 N It is suspended from two vertical springs P and Q. The springs obey Hooke’s law and the spring constant of each spring is 240 N m⁻¹

The top end of spring P is fixed and the top end of spring Q is adjusted until the sign is horizontal and in equilibrium.

What is the extension of spring Q ?

Table 1 shows the results of the experiment.

Show that the data in Table 1 are consistent with the relationship

where T is the tension in the nylon string .

The nylon string used has a density of 1150 kg m^–3 and a uniform diameter of

Determine the length l of the string used.

The student uses the relationship in question 02.1 to predict frequencies for tensions that are much larger than those used in the original experiment.

Explain how the actual frequencies produced would be different from those that the student predicts.