Question

The pointer versus load graph for a spring balance is as shown in the figure below. What is the spring constant?

$$\begin{array}{rl}& A.1.5kgf/cm\\ & B.5kgf/cm\\ & C.\text{ 0}\text{.1}kgf/cm\\ & \text{D}\text{. 10}kgf/cm\end{array}$$

Answer 1

Hint: The extension of a spring balance and the applied load is directly proportional to each other. This idea can be employed in finding the spring constant of the given spring. It is very clear from the graph also that the two quantities are linearly related.

Complete answer:
The force constant or spring constant ‘k’ is derived from the Hooke’s law. The law states that the extension or compression of a spring is directly proportional to the load or force applied on it. The constant of proportionality is the spring constant which is a characteristic property of a spring.
The Hooke’s law is given by:
$$F=-kx$$
Where, F is the force applied,
x is the displacement due to extension or compression,
k is the force constant.
From the diagram given,

It is clear that the extension and the load are linearly related to each other with (0,0) as the origin.
Therefore, we can write this in the straight-line form of $$y=mx+c$$as
$$\begin{array}{rl}& Load,\\ & F=Extension\times k\end{array}$$
$$\Rightarrow F=x\times k$$
The value of F is given by $$(10-0)$$ and x is given by $$(1-0)$$
$$\begin{array}{rl}& \Rightarrow k={\displaystyle \frac{F}{x}}\\ & \Rightarrow k={\displaystyle \frac{10}{1}}=10kgf/cm\end{array}$$
So, we have derived from the graph that the spring constant of the given spring to be
$$10kgf/cm$$.

The answer is option D.

Note:
We can find the force constant just using the slope formula from the graph itself. The Hooke’s law is derived from this graph. The negative sign involved denotes that the extension is in the direction opposite to that of the applied force.