Gravitational Potential Energy: Formula and Derivation

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Gravitational potential energy is the work that a body needs to do, against the force of gravity, in order to bring that body into a particular space. In other words, Gravitational potential energy is the energy that an item acquires owing to a shift of location in the gravitational field.

  • Since energy can neither be created nor destroyed, an object at rest also has some form of energy within it.
  • This energy is known as potential energy. 
  • If the position of the body changes due to external forces, the change in potential energy is proportional to the amount of work done on the body by the forces.
  • Gravitational force is a conservative force as work done is independent of the method employed for a shift in position.

Read More: NCERT Solutions for Class 11 Physics Gravitation

Key Terms:  Gravity, Gravitational Force, Universal Law of Gravitation, Force, Acceleration, Gravitation, Mass.


What is Gravitational Potential Energy?

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If the location of a particle changes due to its force, only the amount of work done on the body by force is the change to its potential energy. Conservative forces are the forces for which the work is autonomous. The strength of gravity is a conservative force and the potential energy of a body that emerges from this force is known as Gravitational Potential Energy.

Gravitation Detailed Video Explanation

  • Individual position defines the potential energy of a body as the energy in that position.
  • Energy shift due to position change of the body by external forces is equal to the amount of work carried out by the forces on the body.
  • The work done by the force of gravity is regardless of the way a change of position takes, in that it might be a conservationist force.
  • Furthermore, all of these forces have potential.
  • Therefore, the gravitational effect in an infinite body is 0 and the potential energy is nil which is known as a reference point.
  • The most frequent use of gravitational potential energy is a gravitational acceleration of approximately 9.8 m/s2 for an object at the surface of the earth.

Read More: Universal Gravitation Formula


Gravitational Potential Energy Formula 

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Gravitational Potential Energy = Mass *Acceleration due to gravity*Height above the ground

GPE = m.g.h

  • Gravitational potential power generates a broad manifestation of the law of gravity, which is equivalent to the effort done to counter gravity by bringing a mass to some point in space.
  • Due to the inverse linear nature of the gravitational force, it is worth choosing the zero of the gravitational energy from an infinite distance.
  • The potential energy of gravity close to a planet is thus negative since gravity works positively as mass approaches.
  • This negative potential shows a "bundled state," which is imprisoned when a mass is near a larger body until it is supplied with enough energy to escape by something.

Read More: Gravitation Important Questions


Gravitational Potential Energy Derivation 

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Due to the source mass ‘M’ initially, a test mass, 'm' is located at an endless point along the x-axis. A little effort is made to achieve this over a very short distance (dx) without acceleration –

dw = Fdx

In this case, F is an attracting force and the shift is in the direction of the negative x-axis so that F is in the same direction. Then,

dw = (GMm/x2)dx

Integrating both sides,

\(\begin{array}{l}w = \int_{\infty }^{r} \frac{GMm}{x^{2}}dx\end{array}\)

\(\begin{array}{l}w = -[\frac{GMm}{x}]_{\infty }^{r}\end{array}\)

\(\begin{array}{l}w = -[\frac{GMm}{r}] – (\frac{-GMm}{\infty })\end{array}\)

\(\begin{array}{l}w = \frac{-GMm}{r}\end{array}\)

As the work is stored as its potential energy U, gravity potential energy is thus produced at a location that is at 'r' distance from the source mass;

U = -GMm/r

  • This formula is useful in calculating escape speed, orbit energy, etc.
  • The acceleration of gravity g can nonetheless be regarded as essentially constant for objects near the Earth.
  • The expression of potential energy relative to the Earth's surface.

U = mgh

Where,

  • h = height above the surface
  • g = surface value of the acceleration of gravity

If a test mass moves inside the same gravitational field as the source mass, the change in the potential energy of the test mass is determined by the location inside the gravitational field;

ΔU = GMm (1/ri – 1/rf)

If ri > rf,

Then ΔU is negative.

Read More: Gravitation MCQs


Gravitational Potential Energy Expression at Height 

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If a body is carried off from the earth's surface to a point 'h' above the earth's surface, then,

ri = rf

And, rf = R+h

ΔU = GMm [1/R – 1/(R+h)]

ΔU = GMmh/R(R + h)

When, h« And, g = GM/R2

Substituting,

The obtained equation is, ΔU = mgh

Check More: Value of g on Moon


Things to Remember

  • The body's weight in the middle of the planet is null because the value of g is zero in the middle of the earth.
  • At a time when the gravitational potential energy is zero, the field of gravity is zero.
  • Gravity Formula is given as F = GM1M2/ R2.
  • The speed of free fall due to gravity is the same over the earth’s surface. This speed on Earth is 9.8 m/s.
  • Weight is the measure of the force of gravity acting on an object or a body.
  • The relation between gravity and weight is given as w = mg.
  • The value of g is calculated to be 9.8 m/s.

Sample Questions

Question. Find the gravitational potential energy of 2.5 kg of weight, maintained at a height of 15 m above the earth. The gravitational force on mass 1 kg is 10 N. (2 marks)

Ans. The formula of Gravitational Potential Energy is, U = mgh

Given , m = 2.5 kg

g = 10 N

h = 15 m

U = 2.5 × 10 × 15

U = 375 J

Ans. 375 J

Ques. The potential energy of a body of mass 0.5 kg increases by 100 J when it is carried to the top of a tower from the ground. If a force of gravity on 1 kg is 10 N, what is the height of the tower? (2 marks)

Ans. We know that, Potential energy = Force × height

U = (mg)h

100 J = (0.5 × 10) N × h

h = 100 / 5

h = 20 m

Ans. 20m

Ques. Calculate the potential gravitational energy of a 10 kg body and is 25 meters above earth. (1 mark)

Ans. Given, m = 10 Kg

h = 25 m

Gravitational Potential Energy, U = mgh

U = 10 * 9.8 * 25

 = 2450 J

Ans. 2450 J

Ques. A rock is raised 10m off the ground and held at an end height of 10m by the person holding it across a 100m field of distance. What has changed the energy potential of gravity? (1 mark)

Ans. The gravity energy change depends solely on the final and beginning height of the item and is regardless of the way to reach the final height. Since the height change for the rock is zero, the energy change for gravity is zero.

Ques. A 2- kg ball hangs from the ceiling of a room with a string length of 1.4 m and a total height of 3.1 metres. About the ceiling, what is the potential ball gravitational energy? Assume 9.8 m/s2 gravity acceleration. (2 marks)

Ans. Gravitational energy is proportional to an object's height and weight.

U = mgh

In our instance this is the basis, the gravitational potential energy is also relative to zero height.

U = 2 * (-9.8) * 1.4

= -27.4 J

Ques. What is the gravitational potential energy? (1 Mark)

Ans. The energy that an item stores as a function of its height or vertical position is known as gravitational potential energy. Because of the Earth's gravitational pull on the item, the energy is captured.

Ques. Why is it called gravitational potential energy? (1 Mark)

Ans. The potential energy that a huge item has in relation to another massive object owing to gravity is known as gravitational energy or gravitational potential energy. When two objects descend toward one another, the potential energy associated with the gravitational field is released.

Ques. What is the expression for gravitational potential energy? (1 Mark)

Ans. Gravitational potential, V = -GMm/r
M is the source mass placed along the x-axis.
m is the test mass at infinity.
r is the distance from the source mass at which the gravitational potential energy is calculated.

Also Read:

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