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(C) The gravitational potential energy of a body of mass $m$ at a distance $r$ from the center of the Earth is given by $U = -\frac{GM_e m}{r}$.Initia

Vedclass · Accepted Answer

$\frac{2}{3} mgR_e$

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(C) The gravitational potential energy of a body of mass $m$ at a distance $r$ from the center of the Earth is given by $U = -\frac{GM_e m}{r}$.Initial potential energy at the surface $(r = R_e)$ is $U_i = -\frac{GM_e m}{R_e}$.Final potential energy at height $h = 2R_e$ $(r = R_e + h = 3R_e)$ is $U_f = -\frac{GM_e m}{3R_e}$.The increase in potential energy is $\Delta U = U_f - U_i$.$\Delta U = -\frac{GM_e m}{3R_e} - (-\frac{GM_e m}{R_e}) = \frac{GM_e m}{R_e} (1 - \frac{1}{3}) = \frac{2}{3} \frac{GM_e m}{R_e}$.Since $g = \frac{GM_e}{R_e^2}$,we have $GM_e = gR_e^2$.Substituting this into the expression: $\Delta U = \frac{2}{3} \frac{(gR_e^2)m}{R_e} = \frac{2}{3} mgR_e$.

Column-$I$	Column-$II$
$(1)$ Can never be positive.	$(a)$ Escape velocity
$(2)$ Reason for negative potential energy of galaxies.	$(b)$ Gravitational potential energy
	$(c)$ The nature of force between different galaxies is attractive.

$A$ body of mass $m$ is taken from the Earth's surface to a height $h$ equal to twice the radius of the Earth $(R_e)$. The increase in potential energy will be: (where $g$ is the acceleration due to gravity on the surface of the Earth)

Similar Questions

$A$ 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 (you may take $G = 6.67 \times 10^{-11}\, Nm^2 / kg^2$).

If the potential at the surface of the Earth is assigned a zero value,then the potential at the center of the Earth will be (Mass $= M$,Radius $= R$).

Write the expression for gravitational potential energy and gravitational potential at the surface of the Earth.

Match Column-$I$ with Column-$II$.
Column-$I$ Column-$II$
$(1)$ Can never be positive. $(a)$ Escape velocity
$(2)$ Reason for negative potential energy of galaxies. $(b)$ Gravitational potential energy
$(c)$ The nature of force between different galaxies is attractive.

If the gravitational potential of an object of mass $m$ on the surface of the Earth is $\phi$,what is the gravitational potential of an object of mass $2m$?

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