Physics > Gravitation > 2.0 Variation of acceleration due to gravity
Gravitation
1.0 Newton's law of gravitation
1.1 Characteristics of gravitational force
1.2 Universal gravitational constant
1.3 Principle of superposition of gravitation
1.4 Gravity
1.5 Acceleration due to gravity
1.6 Relation between $g$ and $G$
2.0 Variation of acceleration due to gravity
2.1 Variation of acceleration due to gravity $(g)$ due to shape of the earth
2.2 Variation of acceleration due to gravity $(g)$ due to altitude
2.3 Variation of acceleration due to gravity $(g)$ due to depth
2.4 Variation of acceleration due to gravity $(g)$ due to rotation of earth
3.0 Gravitational field
3.1 Gravitational field due to a point mass
3.2 Gravitational field due to a uniform solid sphere
3.3 Gravitational field due to a uniform spherical shell
3.4 Gravitatioal field due to a uniform circular ring at a point on its axis
4.0 Gravitational potential
4.1 Gravitational potential due to a point mass
4.2 Gravitational potential due to a uniform solid sphere
4.3 Gravitational potential due to a uniform thin spherical shell
4.4 Gravitational potential due to a uniform ring at a point on its centre
4.5 Relation between gravitational field and gravitational potential
5.0 Gravitational potential energy
5.1 Gravitational potential energy for a system of particles
5.2 Gravitational potential energy of a body on earth's surface
6.0 Satellites
6.1 Orbital speed of satellite
6.2 Time period of a satellite
6.3 Angular momentum of a satellite
6.4 Energy of a satellite
6.5 Types of satellite
6.6 Binding energy
6.7 Escape velocity
6.8 Weightlessness
7.0 Kepler's law of planetary motion
8.0 Problem solving technique
2.3 Variation of acceleration due to gravity $(g)$ due to depth
1.2 Universal gravitational constant
1.3 Principle of superposition of gravitation
1.4 Gravity
1.5 Acceleration due to gravity
1.6 Relation between $g$ and $G$
2.2 Variation of acceleration due to gravity $(g)$ due to altitude
2.3 Variation of acceleration due to gravity $(g)$ due to depth
2.4 Variation of acceleration due to gravity $(g)$ due to rotation of earth
3.2 Gravitational field due to a uniform solid sphere
3.3 Gravitational field due to a uniform spherical shell
3.4 Gravitatioal field due to a uniform circular ring at a point on its axis
4.2 Gravitational potential due to a uniform solid sphere
4.3 Gravitational potential due to a uniform thin spherical shell
4.4 Gravitational potential due to a uniform ring at a point on its centre
4.5 Relation between gravitational field and gravitational potential
5.2 Gravitational potential energy of a body on earth's surface
6.2 Time period of a satellite
6.3 Angular momentum of a satellite
6.4 Energy of a satellite
6.5 Types of satellite
6.6 Binding energy
6.7 Escape velocity
6.8 Weightlessness
Let the particle of mass $m$ is kept at a depth $d$ from the surface of earth. So, the gravitation force exerted on the particle of mass $m$ is,
$$F = \frac{{GM'm}}{{{{\left( {R - d} \right)}^2}}} \quad ...(i)$$
Density of earth is, $$\rho = \frac{M}{{\left( {\frac{4}{3}\pi {R^3}} \right)}}$$
Then mass $M$ is given by, $$M' = \rho V$$$$M' = \frac{M}{{\left( {\frac{4}{3}\pi {R^3}} \right)}} \times \frac{4}{3}\pi {\left( {R - d} \right)^3}$$$$M' = \frac{{M{{\left( {R - d} \right)}^3}}}{{{R^3}}} \quad ...(ii)$$
From equation $(i)$ and $(ii)$ we get,
$$F = \frac{{Gm}}{{{{\left( {R - d} \right)}^2}}} \times \frac{{M{{\left( {R - d} \right)}^3}}}{{{R^3}}}$$$$F = \frac{{GMm\left( {R - d} \right)}}{{{R^3}}}$$$$\frac{F}{m} = \left( {\frac{{GM}}{{{R^2}}}} \right)\left( {\frac{{R - d}}{R}} \right)$$$$g' = g\left[ {1 - \frac{d}{R}} \right]$$
At the centre $(d=R)$ then $g'=0$.
The value of acceleration due to gravity decreases with depth. It is maximum at the earth's surface and minimum at the centre of the earth.
Note:
|
|
| $$F = \frac{{GMm}}{{{{\left( {R + h} \right)}^2}}}$$ | $$F = \frac{{GMm}}{{{R^3}}}\left( {R - d} \right)$$ |
| $$g' = g\left[ {1 - \frac{{2h}}{R}} \right]$$ | $$g' = g\left[ {1 - \frac{d}{R}} \right]$$ |
| If a body is taken above the surface of earth, then the value of acceleration due to gravity varies inversely as the square of the distance from the centre of the earth | If the body is taken inside the earth, acceleration due to gravity decreases linearly with distance from the centre of the earth |
 | |
| $$F = \frac{{GMm}}{{{{\left( {R + h} \right)}^2}}}$$ | $$F = \frac{{GMm}}{{{R^3}}}\left( {R - d} \right)$$ |
| $$g' = g\left[ {1 - \frac{{2h}}{R}} \right]$$ | $$g' = g\left[ {1 - \frac{d}{R}} \right]$$ |
| If a body is taken above the surface of earth, then the value of acceleration due to gravity varies inversely as the square of the distance from the centre of the earth | If the body is taken inside the earth, acceleration due to gravity decreases linearly with distance from the centre of the earth |
