Question

The work done in slowly lifting a body from earth's surface to height r (radius of earth) is equal to two times the work done in lifting the same body from earth's surface to a height h. find the height h.

Answer 1

by formula

mgh
h= 2r .....

Answer 2

Answer:

The height h is $\frac{R}{3}$.

Explanation:

Case I: Lifting to height "r"

$W_1=-\frac{GMm}{r+h}-(-\frac{GMm}{r})$   (1)

Where,

W₁=work done in lifting to height "r"

G=universal gravitational constant

M=mass of the earth

m=mass of the body

r=radius of the earth

h=height to which the body is raised

By placing"h=r" in equation (1) we get;

$W_1=-\frac{GMm}{r+r}-(-\frac{GMm}{r})$

$W_1=-\frac{GMm}{2r}+\frac{GMm}{r}$

$W_1=-\frac{GMm}{2r}+\frac{2GMm}{2r}$

$W_1=\frac{GMm}{2R}$     (2)

Case II: Lifting to height "h"

$W_2=-\frac{GMm}{R+h}-(-\frac{GMm}{R})$

$W_2=-\frac{GMm}{R+h}+\frac{GMm}{R}$

$W_2=\frac{GMmh}{R(R+h)}$     (3)

In the question, it is given that,

$W_1=2W_2$      (4)

By putting W₁ and W₂ in equation (4) we get;

$\frac{GMm}{2R}=2\times \frac{GMmh}{R(R+h)}$

$R+h=4h$

$R=3h$

$h=\frac{R}{3}$ (5)

Hence, the height h is $\frac{R}{3}$.

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