Question

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

Answer 1

The energy of a body due to its position or change in shape is known as potential energy. The energy of a body due to its position above the ground is known as potential gravitational energy.

Potential energy=m x g x h

Where:- m→Mass of the body
             g→Acceleration due to gravity
             h→height of the body above a reference point say the surface of the                         earth.
Please not that Potential Energy usually denoted by the symbol P.E

Answer 2

gravitational force between an object of mass m and the Earth is given by:     F = m g. 
We assume that the force is constant over distance.

Gravitational potential energy is the work done against the Earth's gravitational field when the object is displaced by a unit length ie. ,  1 meter.
 
Suppose the object is displaced by h meters.  (from a height 0 to height h,
     Work done = Force * displacement =   m g *  h
     So this is the potential energy of the mass at height  h.

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We assume that the force (weight) on the object is not a constant over distance.

R = radius of Earth,  h = altitude of object h above Earth's surface.

Newtons laws:      F = G M m / x²

work done =  $\int\limits^d_\infty {F(x)} \, dx \\\\= \int\limits^d_\infty {\frac{GMm}{x^2}} \, dx \\\\=GMm [-\frac{1}{x}]_\infty^d\\\\=-\frac{GMm}{d}$

On the surface of Earth, h = 0, d = R,
              GM/R^2  = g      and  PE = - m g R

at an altitude h:    (h << R)
   Potential energy = - m g R^2 / (R + h)  ≈ - m g  R (1 - h / R)
                   ≈  -  m g R + m g h

 so change in potential energy when the object moves up by a height h
       ≈ m g h