Forces and elasticity

A light rod is hung from a horizontal ceiling using 2 springs, one attached to each end. The springs have force constants k and 2 k.

A downwards force F is applied closer to one end of the rod such that the rod is horizontal.

What is the tension T in the spring with the force constant 2 k ?

The length of an unloaded spring is approximately 4 cm.

The force constant k of the spring is 0.62 N cm^–1.

Describe how you could determine k using an appropriate experiment.

Calculate the magnitude of the block’s initial acceleration at the instant that the force F is removed.

Assume that the spring is not extended beyond its limit of proportionality.

acceleration =........ m s^–2

The block now moves with simple harmonic motion.

Calculate the frequency of this motion.

The block is replaced by a strong magnet L of slightly greater mass.

The oscillation frequency of this new arrangement is 2.5 Hz.

The magnet L is placed inside a coil X of insulated copper wire.

The coil X is connected with long wires to a second, identical coil Y.

A second strong magnet M is placed inside Y and attached to a vibration generator.

The vibration generator is then forced to oscillate with a frequency of approximately 2.5 Hz by

As magnet M oscillates, it moves in and out of coil Y.

........

The frequency of the vibration generator is now varied between 0.5 Hz and 5.0 Hz.

Suggest how the amplitude and frequency of the oscillations of L will change as the frequency of the generator is varied.

You may draw a diagram to support your answer.

Show that the equation above is homogeneous by reducing the equation to SI base units.

Explain why the magnitude of the resultant force F on the glider is given by F = 2 kx where x is the displacement at any time.

State and explain the effect, if any, of increasing the initial displacement on the period of the subsequent motion.

Masses are added to the glider, and the measurement of 20 T repeated.

The results table is below.

Describe two different errors in the table.

Plot the first data point from the table on the graph below.

The other points have all been plotted. The table of results is repeated on the opposite page.

Include on your graph a line of best fit.

Use the graph to determine the value of k.

=........ N m^–1 k

When the initial displacement is increased, one spring increases its extension while the extension of the other spring decreases.

Explain why the maximum kinetic energy of the motion increases.

A spring is stretched by hanging on it a variable mass m. The mass m is always at rest. The spring obeys Hooke’s law.

What is the relationship between the elastic potential energy E in the spring and the mass m ?

A E m^–1

B E m^–2

C E m

D E m²

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.