NIRF Ranking 2024: List of Top Engineering Colleges in India

# Gravitational Potential Energy - Practice Questions & MCQ

Edited By admin | Updated on Sep 18, 2023 18:34 AM | #JEE Main

## Quick Facts

• Gravitational Potential Energy (U) is considered one of the most asked concept.

• 15 Questions around this concept.

## Solve by difficulty

A particle of mass 10 g is kept on the surface of a uniform sphere of mass 100 Kg and radius  10 cm  Find the work to be done against the gravitational force between them to take the particle far away from the sphere.

$\dpi{100} \left ( You\: might\: take\: G= 6.67\times 10^{-11}Nm^{2}/kg^{2} \right )$

Energy required to move a body of mass  from  an orbit of radius  2R  to  3R  is:

## Concepts Covered - 1

Gravitational Potential Energy (U)

It is the amount of work done in bringing a body from  $\infty$  to that point against gravitational force.

• It is Scalar quantity

• SI Unit: Joule

• Dimension : $\left[ ML^{2}T^{-2}\right ]$

• Gravitational Potential energy at a point

If the point mass M is producing the field

Then gravitational force on test mass m at a distance r from M is given by $F=\frac{GMm}{r^2}$

And the amount of work done in bringing a body from $\infty$ to r

= $W=\int_{\infty}^{r}\frac{GMm}{x^2}dx=-\frac{GMm}{r}$

And this is equal to gravitational potential energy

So $U=-\frac{GMm}{r}$

$U \rightarrow$ gravitational potential energy

$M \rightarrow$ Mass of source-body

$m \rightarrow$ mass of test body

$r \rightarrow$ distance between two

Note- U is always negative in the gravitational field because Force is attractive in nature.

Means As the distance r increases U becomes less negative

I.e U will increase as r increases

And for $r=\infty$, U=o which is maximum

• Gravitational Potential energy of discrete distribution of masses

$U=-G\left [ \frac{m_{1}m_{2}}{r_{12}}+\frac{m_{2}m_{3}}{r_{23}}+\cdot \cdot \cdot \right ]$

$U \rightarrow$ Net Gravitational Potential Energy

$r_{12},r_{23}\rightarrow$ The distance of masses from each other

• Change of potential energy

if a body of mass m is moved from  $r_{1 }$ to $r_{2 }$

Then Change of potential energy is given as

$\Delta U=GMm\left [ \frac{1}{r_{1}}-\frac{1}{r_{2}} \right ]$

$\Delta U \rightarrow$ change of energy

$r_{1},r_{2}\rightarrow$ distances

If $r_{1}>r_{2}$ then the change in potential energy of the body will be negative.

I.e To decrease potential energy of a body we have to bring that body closer to the earth.

• The relation between Potential and Potential energy

As $U=\frac{-GMm}{r}=m\left [ \frac{-GM}{r} \right ]$

But $V=-\frac{GM}{r}$

So $U=mV$

Where $V\rightarrow$ Potential

$U\rightarrow$ Potential energy

$r\rightarrow$ distance

• Gravitational Potential Energy at the center of the earth relative to infinity

${U_{centre}=mV_{centre}}\\ {V_{centre}\rightarrow Potential\: at\: centre}$

$U=m\left ( -\frac{3}{2}\frac{GM}{R} \right )$

$m \rightarrow$ mass of body

$M \rightarrow$ Mass of earth

• The gravitational potential energy at height 'h' from the earth's surface

$U_{h}=-\frac{GMm}{R+h}$

Using $GM=gR^2$

$U_{h}=-\frac{gR^{2}m}{R+h}$

$U_{h}=-\frac{mgR}{1+\frac{h}{R}}$

$U_{h}\rightarrow$ The potential energy at the height $h$

$R\rightarrow$ Radius of earth

## Study it with Videos

Gravitational Potential Energy (U)

"Stay in the loop. Receive exam news, study resources, and expert advice!"