Table of Contents

## Gravitational Potential Energy

Gravitational potential energy is the energy moved procured by an item because of an adjustment of its position when it is available in a gravitational field. In straightforward terms, one might say that gravitational potential energy is an energy that is connected with gravitational power or to gravity.

- The most well-known model that can assist you with understanding the idea of gravitational potential energy is assuming you take two pencils.
- One is put at the table and the other is held over the table. Presently, we can express that the pencil which is high will have more prominent gravitational likely energy than the pencil that is at the table.
- What we can realize here is that the pencil or any item, specifically, will possibly take care of business in light of its area in the gravitational field.
- We will find out about the subject exhaustively beneath.

**What is Gravitational Potential Energy?**

Whenever an assortment of mass (m) is moved from endlessness to a point inside the gravitational impact of a source mass (M) without speeding up it, how much work done in uprooting it into the source field is put away as expected energy. This is known as gravitational likely energy. It is addressed with the image Ug.

**Clarification:** We realize that the possible energy of a body at a given position is characterized as the energy put away in the body at that position. In the event that the place of the body changes because of the utilization of outside powers the adjustment of potential energy is equivalent to how much work is done on the body by the powers.

Under the activity of gravitational power, the work done is autonomous of the way taken for an adjustment of position so the power is a moderate power. Also, all such powers have a few potentials in them.

The gravitation effect on a body at limitlessness is zero, consequently, potential energy is zero, which is known as a source of perspective point.

**Gravitational Potential Energy Formula**

The condition for gravitational potential energy is:

⇒ GPE = m⋅g⋅h

Where,

- m is the mass in kilograms
- g is the speed increase because of gravity (9.8 on Earth)
- h is the stature over the ground in meters

## Determination of Gravitational Potential Energy Equation

Consider a source mass ‘M’ is set at a point along the x-pivot, at first, a test mass ‘m’ is at boundlessness. A limited quantity of work done in bringing it without speed increase through a tiny distance (dx) is given by

**dw = Fdx**

Here, F is an appealing power and the relocation is towards the negative x-hub heading so F and dx are in a similar course. Then, at that point,

**dw = (GMm/x ^{2})dx**

Integrating on the two sides

\[w = \int_{\infty }^{r} \frac{GMm}{x^{2}}dx\]

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

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

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

Since the work done is put away as its potential energy U, along these lines gravitational possible energy at a point which is a good ways off ‘r’ from the source mass is given by;

U = – GMm/r

On the off chance that a test mass moves from a point inside the gravitational field to the next point inside a similar gravitational field of source mass, then, at that point, the adjustment of likely energy of the test mass is given by;

**ΔU = GMm (1/r _{i} – 1/r_{f})**

In the event that r_{i} > r_{f}, ΔU is negative.

**Articulation for Gravitational Potential Energy at Height (h) – Derive ΔU = mgh**

Assuming that a body is taken from the outer layer of the earth to a point at a tallness ‘h’ over the outer layer of the earth, then, at that point, r_{i} = R and r_{f} = R + h then, at that point,

ΔU = GMm [1/R – 1/(R+h)]

ΔU = GMmh/R(R + h)

At the point when, h<<R, then, at that point, R + h = R and g = GM/R^{2}.

On subbing this in the above condition we get,

ΔU = mgh

⇒ Note:

- The heaviness of a body at the focal point of the earth is zero because of the way that the worth of g at the focal point of the earth is zero.
- At a point in the gravitational field where the gravitational potential energy is zero, the gravitational field is zero.

**What is Gravitational Potential?**

How much work is done in moving a unit test mass from limitlessness into the gravitational impact of source mass is known as gravitational potential.

Essentially, it is the gravitational potential energy moved by a unit test mass

**⇒ V = U/m**

**⇒ V = – GM/r**

⇒ Significant Points:

- The gravitational potential at a point is generally bad, V is greatest at the vastness.
- The SI unit of gravitational potential is J/Kg.
- The layered equation is M
^{0}L^{2}T^{-2}.

**Dimensional Formula:**

The formula(dimensional) of GPE(Gravitational Potential Energy) is M^{1} L^{2} T^{-2}

Where,

- M = Mass
- L = Length
- T = Time

### Derivation

(G.P.E) = Mass(M) . Acceleration due to gravity . Altitude . . . (1)

- we know that dimensional formula of (M)mass and altitude = [M
^{1}L^{0}T^{0}] and [M^{0}L^{1}T^{0}] . . . . (2) - we know that, the formula(dimensional) of g = [M
^{0}L^{1}T^{-2}] . . . . (3) - by, substituting (2) and (3) in (1) we get,
- Gravitational Potential Energy = Mass × Acceleration due to gravity × Altitude
- So, G.P.E = [M
^{1}L^{2}T^{-2}]. - Therefore, GPE is represented(dimensionally) as
**[M**^{1}L^{2}T^{-2}].

**FAQs**

##### What is implied by gravitational possible energy?

Gravitational potential energy is the energy put away in an item as the aftereffect of its upward position or tallness. The energy is put away as the aftereffect of the gravitational fascination of the Earth for the item.

##### What is the contrast between Potential Energy and Gravitational Potential Energy?

Gravitational potential energy just relies upon the gravitational capability of the point and the mass of the article. Potential energy can rely upon numerous different factors like charge, flow, electric potential and numerous others.