Question

A long spring is stretched by $2cm$ and its potential energy is $U$. If the spring is stretched by $10cm$; its potential energy will be (in terms of $U$)

• A. $U/5$
• B. $U/25$
• C. $5U$
• D. $25U$

Answer 1

The correct option is D $25U$

The potential energy of a stretched spring is
$U=\frac{1}{2}k{x}^2$
Here, $k=$ spring constant,
$x=$ elongation in spring.
But given that, the elongation is $2cm$.
So, $U=\frac{1}{2}k{\left(2\right)}^2$
$\Rightarrow U=\frac{1}{2}k\times 4$ ......(i)
If elongation is $10cm$ then potential energy
$U^{\prime }=\frac{1}{2}k{\left(10\right)}^2$
$U^{\prime }=\frac{1}{2}k\times 100$ ......(ii)
On dividing equation (ii) by equation (i), we have
$\frac{U^{\prime }}{U}=\frac{\frac{1}{2}k\times 100}{\frac{1}{2}k\times 4}$
or $\frac{U^{\prime }}{U}=25\Rightarrow U^{\prime }=25U$