91Ó°ÊÓ

Gravitational force is equal to the restoring force of the spring.

Here spring force is balanced by weight. You can use this fact to find the spring constant. To find the loss of gravitational potential energy, you can use the formula for spring elastic potential energy.

Formula:

Write the equation for the spring force as below.

F = kx

Here, F is the spring force, k is the spring constant, and x is the displacement.

Let x be the displacement of spring, so spring force is given as,

F = kx

This spring force is balanced by weight, so

kx = m x g

Here, m is the mass and g is the acceleration due to gravity.

$⇒k=m\frac{g}{x}$

Now, gain in spring potential energy is given as follows:

$U_e=\frac{1}{2}k{x}^2$

Now plug the value of k in the above equation.

role="math" localid="1661237598418" $$\begin{gathered} U_e=\frac{1}{2}\left(m\frac{g}{x}\right)x^2 \\ {U}_e=\frac{1}{2}mgx............\left(1\right) \end{gathered}$$

But the loss in gravitational potential energy is given as

$U_g=mgx$ ….. (2)

Comparing equations (1) and (2), you obtain

$U_e=\frac{1}{2}{U}_{g}U$

Therefore,

$U_g=2U_e$

Hence, the loss of gravitational potential energy of the cabbage-earth system equals twice the gain in spring’s potential energy is $U_g=2U_e$.