Gravitational Potential (V) Due to Uniform Solid Sphere

Gravitational potential (V) due to uniform solid sphere is an important aspect of Gravitational Potential (V). Gravitational Potential (V) refers to the work done per unit mass in a gravitational field’s point. Some externally applied force is behind the performance of this work. This work would be required to bring a big or massive item to that particular point from a zero potential’s defined position. This usually is the infinity.  Gravitational Potential (V) can also be due to a uniform solid sphere. Keep on reading to understand the concept of gravitational potential (V) due to uniform solid sphere in more detail. Here, we shall take a look at the gravitational potential (V) due to uniform solid sphere: examples and importance.

Gravitational Potential (V) Due to Uniform Solid Sphere: Meaning

The defining of the gravitational potential can take place as the gravitational potential energy per unit mass in a way that is relative to a defined zero potential energy position.

If the Gravitational potential field is due to an object of finite size, then it must be described as the potential to be zero at an infinite distance from this specific object. This gravitational potential is negative everywhere due to the fact that the gravity force has attractiveness.

There is also a possibility of there being a gravitational potential (V) due to a uniform solid sphere. Because of this, the gravitational potential within the sphere is the same.

Gravitational potential due to a Uniform Solid Sphere: Equation

There is a certain amount of work performed for bringing a unit mass from infinity to a gravitational field’s point. At that point, this work is described as the gravitational potential.  

 Now, let us try to find out the gravitational potential (V) due to a uniform solid sphere.

A potential is a scalar field through which the potential energy per unit of a certain quantity can be described. Moreover, this quantity is because of a vector field.

The equation for gravitational potential energy can be expressed as:

⇒ GPE = m⋅g⋅h

Here,

m represents the mass in kilograms,

g represents the gravity acceleration, which is 9.8 on Earth,

h represents the height, in metres, that is above the ground.

Gravitational potential (V) due to uniform solid sphere formula, at a distance r from the centre, is as follows:

• Outside of the sphere V = -GM/r.
• Inside the sphere V = -GM/R3 (R2/2 – r2/6)

Gravitational Potential (V) Due to Uniform Solid Sphere: Examples

Consider that an item of mass M is found at point O. Also, there is a point P at a distance r from O. Now, we need to find out the gravitational potential at point P.

For this point, the gravitational potential would be equal to a unit mass’s potential energy at that point. Now, due to the gravitational force, the work performed is not bounded by the path taken for a positional change. Therefore, we are dealing with a conservative force. Moreover, all such forces are characterised with some potential in them.

For a point mass M at a distance ‘r’, the expression of the gravitational potential is as:

V = – GM/r.

Now, at point P, the gravitational intensity = GM/r2.

Now, the amount of work done can be expressed as:

dV = force x displacement = intensity displacement = (Gm/r2) dr

Therefore, the total work whose performance takes place to bring the unit mass from infinity to point P is expressed as,

V = ∫ dV = r=r∫r=∞ (Gm/r2) dr

or, V = GM r=r∫r=∞ (1/r2.dr) = GM [- 1/r]r∞

or, V = – GM/r

The negative sign is indicative of the fact that no external agent has done the work and that only the gravitational force is involved.

Gravitational Potential (V) Due to Uniform Solid Sphere: Importance

The gravitational potential (V) due to uniform solid sphere: importance can be understood by the following three points.

• By considering all the uniform solid spheres’ masses, the centre potential at the sphere’s outside points can be calculated.
• At infinity, the gravitational influence on an object is zero. As such, potential energy shall also be zero. This way, large spherical bodies in space are able to rotate without any chaos.
• The potential at all the inside solid sphere’s points remains constant. This potential is equal to the sphere’s surface potential. This is why spherical objects like balls can move properly in sports.

Conclusion

The gravitational potential energy per unit mass in a way that is relative to a defined zero potential energy position is termed as gravitational potential. Gravitational potential (V) due to a uniform solid sphere is an important aspect of gravitational potential (V). Due to this, the gravitational potential within the sphere is the same. Its outside of the sphere formula is, V = -GM/r, while its inside the sphere formula is, V = -GM/R3 (R2/2 – r2/6). The examples and importance of gravitational potential (V) due to uniform solid spheres are crucial parts of this topic that need to be studied.