Physics > Gravitation > 4.0 Gravitational potential
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
4.4 Gravitational potential due to a uniform ring at a point on its centre
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
The gravitational potential due to a uniform ring of mass $M$ and radius $R$ at a distance $r$ from the centre is given as,
$$V = - \frac{{GM}}{{\sqrt {{R^2} + {r^2}} }}\quad \quad \left( {0 \leqslant r \leqslant \infty } \right)$$
Proof:
Gravitational field at point $P$ due to circular ring of mass $M$ and radius $R$ is,
$$E = \frac{{GMr}}{{{{\left( {{R^2} + {r^2}} \right)}^{\frac{3}{2}}}}}$$
Similarly, force on mass $m$ at point $P$ is, $$F = Em$$$$F = \frac{{GMmr}}{{{{\left( {{R^2} + {r^2}} \right)}^{\frac{3}{2}}}}}$$
Work done for displacement $dr$ is, $$dW = \overrightarrow F .\,d\overrightarrow r $$$$dW = - \frac{{GMmr}}{{{{\left( {{R^2} + {r^2}} \right)}^{\frac{3}{2}}}}}dr$$
Potential is given by, $$dV = - \frac{{dW}}{m}$$ or $$dV = \frac{{GMrdr}}{{{{\left( {{R^2} + {r^2}} \right)}^{\frac{3}{2}}}}}$$
Integrating with proper limits we get,
$$\int\limits_{{V_\infty }}^{{V_r}} {dV} = \int\limits_\infty ^r {\frac{{GMrdr}}{{{{\left( {{R^2} + {r^2}} \right)}^{\frac{3}{2}}}}}} $$$$\left[ V \right]_{{V_\infty }}^{{V_r}} = GM\left[ { - \frac{1}{{\sqrt {{R^2} + {r^2}} }}} \right]_\infty ^r$$$${V_r} - {V_\infty } = - GM\left( {\frac{1}{{\sqrt {{R^2} + {r^2}} }} - \frac{1}{\infty }} \right)$$ As $\left( {{V_\infty } = 0} \right)$, $${V_r} = - \frac{{GM}}{{\sqrt {{R^2} + {r^2}} }}$$
The $V$ vs $r$ graph can be drawn as,
