Question

Define gravitational potential energy. Deduce an expression for it for a mass in the gravitational field of the Earth.

Answer 1

gravitational potential energy is energy an object possess because of its position in a gravitational field. The most common use of gravitational potential energy is for an object near the surface of the Earth where the gravitational acceleration can be assumed to be constant at about 9.8 m/s2⃣.
GPE is measured in joules, or unit of energy.
GPE = mgh
PEgrav. = mass . g . height

m = mass
h = height
and
g = gravitational field strength.

Answer 2

It is defined as the work done in bringing a body from infinity to that point.

• consider a body of mass m lying at a distance x from earth of mass M

$\displaystyle\sf \:\:\:\:\:\:\:\:\:\;\;\:\implies F = \dfrac{GMm}{x^2}$

If the body is displaced through a distance dx then

$\displaystyle\sf \:\:\:\:\:\:\:\:\:\;\;\:dw = Fdx = \dfrac{GMm}{x^2}\:dx$

Total work done:

$\displaystyle\sf \:\:\:\:\:\:\:\:\:\;\;\:W = \int\limits_{\infty}^r \dfrac{GMm}{r^2}\:dx$

$\displaystyle\sf$

$\displaystyle\sf\:\:\:\:\:\:\:\:\:\;\;\: = GMm \int\limits_{\infty}^r \dfrac{1}{x^2}\:dx$

$\displaystyle\sf$

$\displaystyle\sf \:\:\:\:\:\:\:\:\:\;\;\:= GMm \left| \dfrac{-1}{x}\right|_{\infty}^r$

$\displaystyle\sf$

$\displaystyle\sf \:\:\:\:\:\:\:\:\:\;\;\:= - GMm \left| \dfrac{1}{r} - \dfrac{1}{\infty}\right|$

$\displaystyle\sf$

$\displaystyle\sf \:\:\:\:\:\:\:\:\:\;\;\:= - \dfrac{GMm}{r}$

$\displaystyle\sf$

• This work done is equal to Gravitational Potential Energy.

$\displaystyle\sf$

i.e.,

$\displaystyle\:\:\:\:\:\:\:\:\:\;\;\:\boxed{\bf W = U_g = - \dfrac{GMm}{r}}$