Question

Spring A is stiffer than spring B $\mathit{(}{\mathit{k}}_{\mathit{A}}\mathit{>}{\mathit{k}}_{\mathit{B}}\mathit{)}$. The spring force of which spring does more work if the springs are compressed, (a) the same distance and (b) by the same applied force?

Short answer

1. For the same displacement, the work done by spring A is more than spring B.
2. For the same spring force, the work done by spring B is more than spring A.

Step-by-step solution

The given data

Spring A is stiffer than spring B $(k_A>k_B)$.

Understanding the concept of the spring force

Here, the work done is due to the spring force. So, we have to use a formula for spring force to calculate the work done on the springs.

Formulae:

The force acting on a spring due to stretch,$F=kx$ (i)

The work done on a spring, $W=\frac{1}{2}k{x}^2$ (ii)

Calculation of the work done for same displacement

a)

For the same displacement,

Work done by the springs only depends on the square of the spring constant.

But, it is given that $k_A>k_B$.

Hence, using equation (ii) the work done by the spring A is greater than that of spring B.

Calculation of the work done for same force

b)

For the same spring force,

If spring force is the same, then the spring having greater stiffness compresses less and the spring having less stiffness compresses more.

We are given that spring constant for B is less, so B compresses more simply using equations (i) and (ii), that is,

$$\begin{gathered} k\alpha \frac{1}{x}\mathrm{and} \\ W\alpha \frac{1}{x^2} \end{gathered}$$

Hence, it has more work done than spring A, which compresses less.