Gravitational Potential Energy MCQ - Practice Questions with Answers

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

• It is a Scalar quantity

• SI Unit: Joule

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

• Gravitational Potential energy at a point

             If the point mass M is producing the field

$$
F=\frac{G M m}{r^2}
$$

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

$$
=W=\int_{\infty}^r \frac{G M m}{x^2} d x=-\frac{G M m}{r}
$$
 

         And this is equal to gravitational potential energy

       $\begin{aligned} & \text { So } U=-\frac{G M m}{r} \\ & U \rightarrow \text { gravitational potential energy } \\ & M \rightarrow \text { Mass of source-body } \\ & m \rightarrow \text { mass of test body } \\ & r \rightarrow \text { distance between two }\end{aligned}$

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

  This means As the distance r increases U becomes less negative 

I.e U will increase as r increases

And for , U=o which is maximum

• Gravitational Potential energy of discrete distribution of masses

$$
U=-G\left[\frac{m_1 m_2}{r_{12}}+\frac{m_2 m_3}{r_{23}}+\cdots\right]
$$

$U \rightarrow$ 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

$$
\Delta U=G M m\left[\frac{1}{r_1}-\frac{1}{r_2}\right]
$$

$\Delta U \rightarrow$ change of energy

$$
r_1, r_2 \rightarrow \text { distances }
$$

If $r_1>r_2$ then the change in the potential energy of the body will be negative.
I.e To decrease the potential energy of a body we have to bring that body closer to the earth.

• The relation between Potential and Potential energy

$$
\begin{aligned}
& U=\frac{-G M m}{r}=m\left[\frac{-G M}{r}\right] \\
& \text { As } \\
& \text { But }=-\frac{G M}{r} \\
& \text { So } U=m V
\end{aligned}
$$

Where $V \rightarrow$ Potential
$U \rightarrow$ Potential energy
$r \rightarrow$ distance

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

$\begin{aligned} & U_{\text {centre }}=m V_{\text {centre }} \\ & V_{\text {centre }} \rightarrow \text { Potential at centre } \\ & U=m\left(-\frac{3}{2} \frac{G M}{R}\right) \\ & m \rightarrow \text { mass of body } \\ & M \rightarrow \text { Mass of earth }\end{aligned}$

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

$\begin{aligned} & U_h=-\frac{G M m}{R+h} \\ & \text { Using } G M=g R^2 \\ & U_h=-\frac{g R^2 m}{R+h} \\ & U_h=-\frac{m g R}{1+\frac{h}{R}} \\ & U_h \rightarrow \text { The potential energy at the height } h \\ & R \rightarrow \text { Radius of earth }\end{aligned}$