13.6 Work by spring force  (Page 2/2)

Work by spring force

Now, we are set to obtain an expression for the work done by the spring. We give the block a jerk to the right as discussed earlier. Let the extension of the spring be “x”. Then,

$$\begin{array}{l}F=-kx\end{array}$$

As the force is variable, we can not use the expression “Frcosθ” to determine work. It is valid for constant force only. We need to apply expression, involving integration to determine work :

$$\begin{array}{l}{W}_S=\int F\left(x\right)dx\end{array}$$

Let $x_i$ and $x_f$ be the initial and final positions of the block with respect to origin, then

$$\begin{array}{l}{W}_S={\int }_{x_i}^{x_f}-kxdx\\ \Rightarrow W_S=-k[\frac{x^2}{2}{]}_{x_i}^{x_f}\\ \Rightarrow W_S=\frac{1}{2}k({x_i}^2-{x_f}^2)\end{array}$$

If block is at origin, then $x_i=0$ and $x_f=x$ (say), then

$$\begin{array}{l}\Rightarrow W_S=-\frac{1}{2}k{x}^2\end{array}$$

When block is subjected to the single force due to spring, the work is given by above expressions. During motion towards right, the spring force on the block acts opposite to the direction of motion. Its magnitude increases with increasing displacement. Thus, spring force does negative work transferring energy "from" the block. As a consequence, kinetic energy of the block (speed) decreases.

This process continues till the velocity and kinetic energy of the block are zero. Spring force, then, pulls the block towards the mean position i.e. origin in the negative x - direction. The spring force, now, is in the direction of motion. It does the positive work on the block transferring energy to the block.

The process continues till the particle returns to the initial position, when its velocity is same as that in the beginning. As there is no dissipative force like friction, kinetic energy of the block on return, at mean position i.e. origin, is equal to that in the beginning.

$$\begin{array}{l}\Rightarrow K_f=K_i\end{array}$$

For this round trip, net work by spring force is zero. Net transfer of energy “to” or “from” the particle is zero. Initial kinetic energy of the particle is retained at the end of round trip. Thus, we can see that the motion under spring is lot similar as that of motion under gravity.

Can we guess here - what will happen to block hereafter? The block has velocity towards left. As such, it will move past the origin. However, spring force will come into picture immediately with compression in the spring and pull the block in the opposite direction i.e towards origin. In doing so, the spring force draws kinetic energy "from" the system. The process continues till the block stops towards an extreme position on the left.

Thereafter, the spring force accelerates the block in the opposite direction i.e. towards origin. If there is no dissipation of energy involved (an ideal condition), this process continues and the block oscillates about the origin.

Motion with spring and other force(s)

We must understand that work by spring force, for a given displacement, is independent of the presence of other forces. The work done by spring remains same.

For this situation, “work-kinetic energy” theorem has following form :

$$\begin{array}{l}\Rightarrow K_f-K_i=W_S+W_F\end{array}$$

where $W_S$ and $W_F$ are the work done by the spring force and other applied force(s) respectively.

Here, work done by other external force(s) may be analyzed with respect to following different conditions :

• Initial and final speeds are zero.
• Initial and final speeds are same.
• Initial and final speeds are different.

In the first two cases, initial and final kinetic energy are same. Hence,

$$\begin{array}{l}\Rightarrow K_f-K_i=W_S+W_F=0\\ \Rightarrow W_F=-W_S\end{array}$$

This is an important result. This means that we can simply compute the work done by spring force and assign the same preceded by a negative sign as the work done by other force(s). Such situation can arise when external force displaces the block and extends the spring such that end velocities are either zero or same.

In third case, kinetic energies at end points are not same. However, work done by spring force remains same as before. Thus, the difference in kinetic energy during a motion is attributed to the net work as done by spring and other forces.