Question

what is gravitational potential energy? also derive it.​

Answer 1

Answer:

Potential energy-

- It is the energy by virtue of an object's position relative to other objects.

- Potential energy is often associated with restoring forces such as a spring or the force of gravity.

● Expression-

Consider an object of mass m raised above height h from earths surface, work done here is given by

Work = force × displacement

W = F.h

W = mgh

But work done here is nothing but potential energy.

PE = mgh

Explanation:

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Answer 2

Explanation:

Gravitational potential energy is the energy possessed or acquired by an object due to a change in its position when it is present in a gravitational field. In simple terms, it can be said that gravitational potential energy is an energy which is related to gravitational force or to gravity.

The most common example that can help you understand the concept of gravitational potential energy is if you take two pencils. One is placed at the table and the other is held above the table. Now, we can state that the pencil which is high will have greater gravitational potential energy that the pencil that is at the table.

Consider a source mass ‘M’ is placed at a point along the x-axis, initially, a test mass ‘m’ is at infinity. A small amount of work done in bringing it without acceleration through a very small distance (dx) is given by

dw = Fdx

Here, F is an attractive force and the displacement is towards the negative x-axis direction so F and dx are in the same direction. Then,

dw = (GMm/x2)dx

Integrating on both sides

w = \int_{\infty }^{r} \frac{GMm}{x^{2}}dxw=∫

∞

r

x

2

GMm

dx

w = -[\frac{GMm}{x}]_{\infty }^{r}w=−[

x

GMm

]

∞

r

w = -[\frac{GMm}{r}] – (\frac{-GMm}{\infty })w=−[

r

GMm

]–(

∞

−GMm

)

w = \frac{-GMm}{r}w=

r

−GMm

Since this work done is stored as its potential energy U, therefore gravitational potential energy at a point which is at a distance ‘r’ from the source mass is given by;

U = -GMm/r

If a test mass moves from a point inside the gravitational field to the other point inside the same gravitational field of source mass, then the change in potential energy of the test mass is given by;

ΔU = GMm (1/ri – 1/rf)

If ri > rf then ΔU is negative.