Physics Assignment Help With Gravitational Field

6.2 Gravitational Field

It is the region around a mass upto which another mass can experience its influence.

Gravitational field is described by the two quantities as shown in diagram given below :

(a) Gravitational field intensity

(b)Gravitational potential

gravitational field

6.2.1 Gravitational Field Intensity

The intensity of gravitational field at a point is the force acting per unit mass of a test particle kept at that point i.e.gravitational field intensity . The direction of gravitational field intensity is same as that of gravitational field intensity.

Since intensity of gravitational field

\ intensity of gravitational field

i.e. intensity of gravitational field source mass

It is a unit vector whose direction is always towards the source mass.

6.2.2 Field Due to Uniform Thin Spherical Shell

Consider a thin spherical shell or radius R, mass M and of negligible thickness. Out of the spherical shell we consider a small ring of thickness Rdq. The shaded ring has mass dm=(M/2)sinq dq. The field at p due to this ring is

spherical shell

From DOAP

z2 = a2 + r2 – 2ar cosq

or 2zdz = 2ar sinq dq

or sinq dq = zdz / a.r

Also, from DOAP,

a2 = z2 + r2 – 2zr. cosa
spherical shell

spherical shell

Thus, spherical shell

or spherical shell

Case I (p is outside the shell, r > a)

spherical shell

We see that the shell may be treated as a point particle of the same mass placed at its centre to calculate the gravitational field at an external point.

(inside the shell, r < a).

spherical shell

We see that field inside a uniform spherical shell is zero.

6.2.3 Relation between gravitational field and potential

Suppose the gravitation field at a point gravitation field due to a given mass distributed is mass distributed. By definition the force on a particle of mass m when it is at distancegravitation field .

potential

As the particle is displaced from gravitational force the work done by the gravitational force on it is

gravitational force

The change in potential energy during this displacement is

potential energy

The change in potential is, by equation

change in potential... (1)

If we work in Cartesian coordinates, we can write

cartesian coordinates

and cartesian coordinates

So that cartesian coordinates

Equation (1) may be written as

dV = – ex dx – ey dy – Ez dz

If y and z remain constant, dy = dz = 0

Thus, gravitational potential

Similarly, gravitational potential and gravitational potential

The symbol gravitational potential means partial differentiation with respect to x treating y and z to be constants.

Gravitational Potential

It is defined as the work done in moving unit mass from infinity to that point against the field. If W work is done in moving a mass m then V = W/m.

Its units are Joule/kg. It is a scalar quantity. its proper sign is negative. In integral from gravitational potential is given by gravitational potential. The gravitational potential on the surface of a sphere of mass M & radius R is, V = – GM / R

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