Gravitational Potential Energy MCQ - Practice Questions with Answers

It is the amount of work done in bringing a body from  $\mathrm{\infty }$  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

$$F=\frac{GMm}{r^2}$$

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

$$=W={\int }_{\mathrm{\infty }}^r\frac{GMm}{x^2}dx=-\frac{GMm}{r}$$
 

         And this is equal to gravitational potential energy

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

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

$$U=-G[\frac{m_1{m}_2}{r_{12}}+\frac{m_2{m}_3}{r_{23}}+\cdots ]$$

$U\to$ Net Gravitational Potential Energy
$r_{12},r_{23}\to$ 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

$$\mathrm{\Delta }U=GMm[\frac{1}{r_1}-\frac{1}{r_2}]$$

$\mathrm{\Delta }U\to$ change of energy

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

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

$$\begin{array}{rl}& U=\frac{-GMm}{r}=m\left[\frac{-GM}{r}\right]\\ & \text{ As }\\ & \text{ But }=-\frac{GM}{r}\\ & \text{ So }U=mV\end{array}$$

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

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

$\begin{array}{rl}& U_{\text{centre }}=m{V}_{\text{centre }}\\ & V_{\text{centre }}\to \text{ Potential at centre }\\ & U=m(-\frac{3}{2}\frac{GM}{R})\\ & m\to \text{ mass of body }\\ & M\to \text{ Mass of earth }\end{array}$

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

$\begin{array}{rl}& U_h=-\frac{GMm}{R+h}\\ & \text{ Using }GM=g{R}^2\\ & U_h=-\frac{g{R}^{2}m}{R+h}\\ & U_h=-\frac{mgR}{1+\frac{h}{R}}\\ & U_h\to \text{ The potential energy at the height }h\\ & R\to \text{ Radius of earth }\end{array}$