Question

Prove that at points near the surface of the Earth, the gravitational potential energy of the object is U = mgh.

Answer 1

When an object of mass m is raised to a height h, the potential energy stored in the object is mgh. This can be derived using the general expression for gravitational potential energy.

Mass placed at a distance r from the center of the Earth

Consider the Earth and mass system, with r, the distance between the mass m and the Earth’s center. Then the gravitational potential energy.

U = `-("GM"_"e""m")/"r"` ....(1)

Here r = R_e + h, where R_e is the radius of the Earth, h is the height above the Earth’s surface

U = `-"G"("M"_"e""m")/("R"_"e" + "h")` ....(2)

If h << Re , equation (2) can be modified as

U = `-"G"("M"_"e""m")/("R"_"e"(1 + "h"/"R"_"e"))`

U = `-"G"("M"_"e""m")/("R"_"e")(1 + "h"/"R"_"e")^-1` ...........(3)

By using Binomial expansion and neglecting the higher order terms, we get

`(1 + "x")^"n" = 1 + "nx" + ("n"("n" - 1))/(2!) "x"^2 + ... + ∞`

Here, x = `"h"/"R"_"e"` and n = − 1

`(1 + "h"/"R"_"e")^-1 = (1 - "h"/"R"_"e")`

Replace this value and we get,

U = `-("GM"_"e""m")/"R"_"e" (1 - "h"/"R"_"e")` .....(4)

We know that, for a mass m on the Earth’s surface,

`"G"("M"_"e""m")/"R"_"e" = "mgR"_"e"`  .....(5)

Substituting equation (4) and (5) we get,

U = −mg_e + mgh ……..(6)

It is clear that the first term in the above expression is independent of the height h. For example, if the object is taken from height h₁ to h₂, then the potential energy at h₁ is

U(h₁) = – mgR_e + mgh₁ …(7)

and the potential energy at h₂ is

U(h₂) = – mgR_e + mgh₂ …(8)

The potential energy difference between h₁ and h₂ is

U(h₂) – U(h₁) = mg(h₂ – h₁) …(9)

The term mgR_e in equations (7) and (8) plays no role in the result.

Hence in equation (6) the first term can be omitted or taken to zero.

Thus it can be stated that the gravitational potential energy stored in the particle of mass m at a height h from the surface of the Earth is U = mgh.

On the surface of the Earth, U = 0, since h is zero.