Work Done, Elasticity and Hooke's Law
Work Done
When a force causes an object to move, energy is transferred. The amount of energy transferred is called the work done. Work done is calculated using:
$$W = F s$$
Where \(W\) is work done in joules (J), \(F\) is force in newtons (N), and \(s\) is the distance moved along the line of action of the force in metres (m).
One joule of work is done when a force of one newton moves an object one metre in the direction of the force. Work done equals energy transferred: \(W = E\).
If a force acts at right angles to the direction of motion, no work is done. For example, gravity does no work on a satellite in a circular orbit because the force is perpendicular to the direction of travel.
A box is pushed 8 m across a floor against a friction force of 25 N. Calculate the work done against friction.
\(W = Fs = 25 \times 8 = 200\) J
This 200 J is transferred to the thermal energy store of the box and floor — they warm up slightly.
Hooke’s Law
Hooke’s law states that the extension of an elastic object is directly proportional to the force applied, as long as the limit of proportionality is not exceeded.
$$F = k e$$
Where \(F\) is force in newtons, \(k\) is the spring constant in N/m, and \(e\) is the extension in metres.
A force-extension graph for a spring is a straight line through the origin up to the limit of proportionality. Beyond this point, the graph curves and the spring no longer returns to its original length when the force is removed — it has been permanently deformed.
Hang a spring from a clamp stand. Measure the original (natural) length. Add masses one at a time and record the new length each time. Calculate the extension by subtracting the original length from the stretched length.
Plot a force-extension graph. The gradient of the straight-line section gives the spring constant \(k\). Use a ruler with a pointer (set square) to reduce parallax error when measuring length.
Elastic Potential Energy
When a spring is stretched or compressed within its limit of proportionality, it stores elastic potential energy. This can be calculated using:
$$E_e = \frac{1}{2} k e^2$$
Where \(E_e\) is elastic potential energy in joules, \(k\) is the spring constant in N/m, and \(e\) is the extension in metres.
A spring with a spring constant of 40 N/m is stretched by 0.15 m. Calculate the elastic potential energy stored.
\(E_e = \frac{1}{2} k e^2 = \frac{1}{2} \times 40 \times 0.15^2 = \frac{1}{2} \times 40 \times 0.0225 = 0.45\) J
An object that returns to its original shape after the force is removed is called elastic. An object that is permanently deformed is said to have undergone inelastic deformation.