Elastic strain energy & plastic deformation

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.

The 500 m tall Taipei 101 tower is shown in Fig. 2.1. The tower has a massive sphere suspended across five floors near the top of the building to dampen down movement of the tower in high winds and earthquakes. The sphere is connected to pistons (not shown) which drive oil through small holes providing damping. The vibration energy of the sphere is converted to thermal energy.

Fig. 2.2 models the damper system as the sphere held between two springs. The movement of the walls of the tower forces the sphere to oscillate in simple harmonic motion.

In the strongest wind, the natural frequency of the oscillations of the tower is 0.15 Hz and the maximum acceleration of the sphere is 0.050 m s⁻².

Calculate the maximum displacement of the sphere in the strongest wind.

Explain why the natural frequency of the damper system must be about 0.15 Hz.

The acceleration a of the sphere is given by the equation

where k is the force constant of the spring combination, x is the displacement of the sphere and m is the mass of the sphere.

The mass of the sphere is 6.6 × 10⁵ kg. The natural frequency of the oscillations of the sphere is 0.15 Hz.

The S-wave of an earthquake causes a sudden movement of the building displacing the sphere 0.71 m from its equilibrium position relative to the building.

Use your answer in (i) to calculate the energy transferred to the springs of the damper system.

A tensile force of 4.5 N is applied to a spring. The spring extends elastically by 3.2 cm.

What is the elastic potential energy of the spring?

A 0.072 J

B 0.14 J

C 2.4 J

D 14 J

Your answer