Question

Two springs of spring constant $1500N/m$and 3000N /m respectively are stretched with the same force. They will have the potential energies in the ratio of 

(A) $1:9$

(B) $1:4$

(C) $2:1$

(D) $4:1$

Answer 1

Hint: The resistive force developed in the spring, when its length is changed it is called a spring force. Potential energy of a spring is the energy associated with the state of compression or expansion of an elastic spring. Spring force is usually given by $F{\text{ }} =  - kx$. Here $k$ is the spring constant and $x$ is the displacement in the spring. 

Complete step by step solution

When work is done in pulling the string, it gets stored as potential energy (U). 

$\Rightarrow U = \dfrac{1}{2}\mathop {kx}\nolimits^2 … (1)$

The restoring force developed in spring is directly proportional to elongation or compression $x$ 

$\Rightarrow F{\text{ }} = {\text{ }}kx{\text{ }} … (2)$

Now, $x = \dfrac{F}{k}$ 

Putting this value in equation (1) 

$\Rightarrow U = \dfrac{1}{2}\mathop {k\left( {\dfrac{F}{k}} \right)}\nolimits^2 $

Now,  $\dfrac{{\mathop U\nolimits_1 }}{{\mathop U\nolimits_2 }} = \dfrac{{\mathop k\nolimits_2 }}{{\mathop k\nolimits_1 }}$ 

Now putting values of spring constants-  

$\mathop K\nolimits_1 {\text{ }} = {\text{ }}1500{\text{ }}N/m$

$\mathop K\nolimits_2 {\text{ }} = {\text{ 30}}00{\text{ }}N/m$ 

$ \Rightarrow \dfrac{{3000}}{{1500}} = \dfrac{2}{1} = 2:1$ 

Therefore, Option (C) is correct.

Note: The energy possessed by a body or system by virtue of its position or configuration is known as the potential energy. For example, a block attached to a compressed or elongated spring possesses some energy called elastic potential energy. This block has a capacity to do work.