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.