Gravitational Potential Energy - Practice Questions & MCQ

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

• It is 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

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

$
\mathrm{So}_{\mathrm{So}}=-\frac{G M m}{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 $r=\infty, \mathrm{U}=\mathrm{o}$ 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
$r_1, r_2 \rightarrow$ 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

$
U=\frac{-G M m}{r}=m\left[\frac{-G M}{r}\right]
$

But $V=-\frac{G M}{r}$
So $U=m V$
Where $V \rightarrow$ Potential
$U \rightarrow$ Potential energy
$r \rightarrow$ distance

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

  $
  \begin{gathered}
  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{gathered}
  $

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

  $
  U_h=-\frac{G M m}{R+h}
  $

  
  Using $G M=g R^2$

  $
  \begin{aligned}
  U_h & =-\frac{g R^2 m}{R+h} \\
  U_h & =-\frac{m g R}{1+\frac{h}{R}}
  \end{aligned}
  $

  $U_h \rightarrow$ The potential energy at the height $h$
  $R \rightarrow$ Radius of earth