Gravitational Potential Energy

Gravitational Potential Energy

This is the amount of work to be done in bringing a body from its reference position to a particular position with under gravity is called gravitational potential energy. Reference position is taken as zero potential at the earth surface, then the GPE at height h is, U = mgh. Position for zero potential energy is infinity.

Suppose a particle having very small mass m is brought to point P from infinity which is at a distance x from earths center. Gravitational force on the particle F = G M_e R_e /x²

This force of attraction brings any mass towards the center of the earth. For a small particle distance dx towards centre of earth.

dW = F dx = GM_em/x² dx

The total work done from infinity to P (at a distance r from the centre of earth) is the sum of infinite small work done given as 

Thus the work is stored as the gravitational potential energy (u) of the particle.

U = -GM_em/r

If height from the earth’s centre is h then work done is,

W = -U = GM_em/r = GM_em/(R_e + h)

If the particle is moved from earth’s surface to height h then,

W= change in potential energy of a particle

= U₂ – U₁

= GM_em/R_e + h - (-GM_em/R_e)

= GM_emh/{R_e (R_e + h)}

The acceleration because of the gravity is given as.

g = GM_e / R_e²

W = Δu = R_e²gmh/ R_e (R_e + h) = R_egmh/R_e + h

W = Δu = mgh/{1+h/R_e}

Then whenever the height h << R_e then the,

W = Δu = mgh.

Gravitational Potential Energy Negative:

If distance between two bodies changes, the gravitational force between them does some work either, resulting in a change in gravitational potential energy of bodies.

The negative sign of the gravitational potential energy indicated that the potential energy is due to the gravitational attraction between the earth and the body. When the body is brought from infinity, the force of gravity does positive work, hence GPE decreases.