Question

Obtain dimensions of force constant ‘k’.

Answer 1

Hint: As a very first step, what we have to do is that we should define the given quantity that is the force constant ‘k’. For that, you could get help from Hooke's law and then get the mathematical expression for the same. Now, you could simply take dimensions on both sides and hence get the answer.
Formula used:
Hooke’s law,
$F=-kx$

Complete answer:
In the question we are asked to obtain the dimensions of the force constant ‘k’. Before going into the dimension, let us first recall the definition of this quantity.
For defining this quantity we have to look into the Hooke’s law. Hooke’s law states that the compressive/expansive force is directly proportional to the distance a string gets stretched. Mathematically,
$F\propto x$
$\Rightarrow F=-kx$
So we see that the force constant which is also known as spring constant is the proportionality constant in this expression.
So, we have got the expression for force constant as,
$k=-\dfrac{F}{x}$……………………………………….. (1)
Now let us recall the dimensions of each quantity.
The dimension of restoring force F would be,
$\left[ F \right]=\left[ ML{{T}^{-2}} \right]$
Dimension of distance x would be,
$\left[ x \right]=\left[ L \right]$
Taking dimensions on both sides of equation (1) we get,
$\left[ k \right]=\dfrac{\left[ ML{{T}^{-2}} \right]}{\left[ L \right]}$
$\therefore \left[ k \right]=\left[ M{{T}^{-2}} \right]$
Therefore, we found the dimension of the force constant to be $\left[ M{{T}^{-2}} \right]$.

Note:
In the expression for Hooke’s law, F is the restoring force present in the string that has direction towards the equilibrium position, k would be the spring/force constant and x would be the displacement from equilibrium position. The negative sign in the expression simply is an implication that the force and displacement are directed opposite to each other. As there are no dimensions for constants (here, -1) we shouldn’t worry about it while finding the dimension of the force constant.