13.8 Conservative force  (Page 3/3)

Problem : A small block of 0.1 kg is released from a height 5 m as shown in the figure. The block following a curved path tansitions to a linear horizontal path and hits the spring fixed to a wedge. If no friction is involved and spring constant is 1000 N/m, find the maximum compression of the spring.

Motion along closed path

Solution : Here we shall make use of the fact that only conservative force is in play. We need to know the speed of the block in the horizontal section of the motion, before it strikes the spring. We can use "work - kinetic energy" theorem. But, the component of gravity along the curved path is a variable force. It would be difficult to evaluate the work done in this section, using integration form of formula for work.

Knowing that work done is independent of the path, we can evaluate the workdone for motion along a vertical straight path.

$$\begin{array}{l}W=mgh\end{array}$$

According "work - kinetic energy" theorem,

$$\begin{array}{l}{K}_f-K_i=W\end{array}$$

Here, speed at the time of release is zero. Thus, $K_i$ is zero. Let the speed of the block on the horizontal section before hitting the spring be "v", Then,

$$\begin{array}{l}\Rightarrow \frac{1}{2}m{v}^2=mgh\end{array}$$

Now, we switch our consideration to the compression of the spring. Let the maximum compression be "x". Then according "work - kinetic energy" theorem,

$$\begin{array}{l}{K}_f-K_i=W\end{array}$$

Here, speed at maximum compression is zero. Thus, $K_f$ is zero.

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

Combining equations :

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

$$\begin{array}{l}\Rightarrow x=\surd \left(\frac{2mgh}{k}\right)\end{array}$$

$$\begin{array}{l}\Rightarrow x=\surd \left(\frac{2x0.1x5x5}{1000}\right)=0.1m=10\mathrm{cm}\end{array}$$

Note : This problem can be solved with simpler calculation using the concept of the conservation of mechanical energy. Energy concept will eliminate intermediate steps involving "work-kinetic energy" theorem. This point will be elaborated in the module on conservation of mechanical energy.