If  g  is the acceleration due to gravity on the earth's surface,  the gain in the potential energy of an object of mass  m   raised from the surface of the earth to   a height equal to the radius  R   of the earth is :

  • Option 1)

    2mgR

  • Option 2)

    \frac{1}{2}mgR

  • Option 3)

    \frac{1}{4}mgR

  • Option 4)

    mgR

 

Answers (1)

As we learnt in 

Force \; on \; object\; =\frac{GMm}{x^{2}}\; at\: x\; from \; centre \; of \; earth.

\therefore \; \; \; Work\; done=\frac{GMm}{x^{2}}dx

\therefore \; \; \int Work \; done=GMm\int_{R}^{2R}\frac{dx}{x^{2}}

\therefore \; \; \; Potential \; energy\; gained

=GMm\left [ -\frac{1}{x} \right ]_{R}^{2R}=\frac{GMm\times 1}{2R}

\therefore \; \; Gain \; in \; P.E.

=\frac{1}{2}mR\left ( \frac{GM}{R^{2}} \right )=\frac{1}{2}mgR\; \; \;\; \; \; \; \; \; \; \left [ \because g=\frac{GM}{R^{2}} \right ]

 

Work done against gravity when 'h' is not negligible -

W=\frac{mgh}{1+\frac{h}{R}}

W=work done

h\rightarrow height above surface of earth

R\rightarrow Radius of earth

- wherein

if h=R

W=\frac{1}{2}mgR

if h=nR

W=mgR\left ( \frac{n}{n+1} \right )

n\rightarrow times

n=1,2,3\cdot \cdot \cdot

 

F = \frac{Gm_{m}}{x^2} at \: x\ from\ centre\ of\ earth

dw = \frac{Gm_{m}}{x^2}dx = \int dx = Gm_{m} \int_{2R}^{R} \frac{dx}{x^2}

\therefore Potential\ energy\ gained = Gm_{m}|-\frac{1}{x}|

= \frac{Gm_{m}\times 1}{2R}

\therefore Gain\ in\ potential\ energy = \frac{1}{2}mR \left ( \frac{Gm}{R^2} \right )

= \frac{1}{2}mgR

 


Option 1)

2mgR

Incorrect

Option 2)

\frac{1}{2}mgR

Correct

Option 3)

\frac{1}{4}mgR

Incorrect

Option 4)

mgR

Incorrect

Preparation Products

JEE Main Rank Booster 2021

This course will help student to be better prepared and study in the right direction for JEE Main..

₹ 13999/- ₹ 9999/-
Buy Now
Knockout JEE Main April 2021 (Subscription)

An exhaustive E-learning program for the complete preparation of JEE Main..

₹ 4999/-
Buy Now
Knockout JEE Main April 2021

An exhaustive E-learning program for the complete preparation of JEE Main..

₹ 22999/- ₹ 14999/-
Buy Now
Knockout JEE Main April 2022

An exhaustive E-learning program for the complete preparation of JEE Main..

₹ 34999/- ₹ 24999/-
Buy Now
Knockout JEE Main January 2022

An exhaustive E-learning program for the complete preparation of JEE Main..

₹ 34999/- ₹ 24999/-
Buy Now
Exams
Articles
Questions