Question

You have two springs that are identical except that spring 1 is stiffer than spring 2 \(\left( {{k_{\bf{1}}}{\bf{ > }}{k_{\bf{2}}}} \right)\). On which spring is more work done: (a) if they are stretched using the same force; (b) if they are stretched the same distance?

Short answer

(a) If both springs are stretched using the same force, more work is done on spring 2.

(b) If both springs are stretched by the same distance, more work is done on spring 1.

Step-by-step solution

Meaning of spring force

Spring force can be defined as the amount of force exerted by an elongated or compressed spring on any object connected to the spring.

Application of Hooke’s law; if both the springs are stretched using the same force

(a)

According to Hooke’s law, the expression for the spring force is given by:

\(F = - kx\)

Here, \(k\) is the spring stiffness and \(x\) is the compression of the spring.

If both springs are stretched using the same force, then according to the proportionality rule, you can write:

\(\begin{aligned}{k_1}{x_1} &= {k_2}{x_2}\\\frac{{{k_1}}}{{{k_2}}} &= \frac{{{x_2}}}{{{x_1}}}\end{aligned}\)

It is given that \(\left( {{k_1} > {k_2}} \right)\). Therefore, \({x_2} > {x_1}\).

Work is defined as the product of force and displacement (compression/elongation). Since the force is of equal magnitude and the displacement for spring 2 is more than spring 1, more work is done on spring 2.

Application of Hooke’s law; if the same distance stretches both the springs

(b)

The spring force is given as follows:

\(F = kx\)

Since the same distance stretches both springs, so the forces in both springs are provided as:

\(\begin{aligned}{F_1} &= {k_1}x\\{F_2} &= {k_2}x\end{aligned}\)

It is given that \(\left( {{k_1} > {k_2}} \right)\). Therefore, \({F_1} > {F_2}\).

Work is defined as the product of force and displacement (compression/elongation). Since the displacement is of equal magnitude and the spring force for spring 1 is more than the spring force for spring 2, more work is done on spring 1.