Question

Two bodies of equal masses are placed at heights h and 2h. The ratio of their gravitational potential energies is:
A. 1
B. 2
C. $\dfrac{1}{2}$
D. $\dfrac{1}{4}$

Answer 1

Hint: $U=mgh$ is the relation of gravitational potential energy and the height at which the object is placed.

Complete step by step answer:
The formula of potential energy is;
$U=mgh$, where m is the object, g is the acceleration due to gravity and h is the height at which the object is placed.
Consider two objects of the same mass and places at different heights.
So ${{h}_{1}}$ is the height at which the first body is placed and ${{h}_{2}}$ is the height at which the second body placed.
From this date we can find the potential energy of each body.
Potential energy of first body, ${{U}_{1}}=mg{{h}_{1}}$
Potential energy of second body, ${{U}_{2}}=mg{{h}_{2}}$
Ratio of the potential energy of these two objects will be, $\dfrac{{{U}_{1}}}{{{U}_{2}}}=\dfrac{mg{{h}_{1}}}{mg{{h}_{2}}}$
Here, m and g are constant for two bodies. Thus, $\dfrac{{{U}_{1}}}{{{U}_{2}}}=\dfrac{{{h}_{1}}}{{{h}_{2}}}$
We can assign values to the heights,
As per the given data, ${{h}_{1}}=h$ and ${{h}_{2}}=2h$.
$\dfrac{{{U}_{1}}}{{{U}_{2}}}=\dfrac{h}{2h}$
$\dfrac{{{U}_{1}}}{{{U}_{2}}}=\dfrac{1}{2}$
Hence the ratio of potential energy is $1:2$. Therefore, the option C is correct.

Additional information:
Potential energy is a form of energy by the virtue of the shape or position of the body. Stretched bows, stretched rubber bands are coming under the potential energy. Similarly, an object placed at a height has the potential energy to do work. For this energy gravity of earth is also responsible. It is actually formulated from Newton’s second law of motion.
$\text{Workdone = force }\times \text{ displacement}$
$\text{Force = mass }\\!\\!\times\\!\\!\text{ acceleration}$
$F=mg$, where g is the acceleration due to gravity.
Displacement = h
Therefore, $\text{Workdone = mgh}$
This work done is actually the energy gained by the object and it is known as gravitational potential energy.

Note: The potential energy will increase with the increase of height, since direct relationship of height and potential energy.