8.18 Summary

• Newton’s law of universal gravitation states that the gravitational force of attraction between any two particles of masses \(m_1\) and \(m_2\) separated by a distance \(r\) has the magnitude
  \(
  F=G \frac{m_1 m_2}{r^2}
  \)
  where \(G\) is the universal gravitational constant, which has the value \(6.672 \times 10^{-11} \mathrm{~N} \mathrm{~m}^2 \mathrm{~kg}^{-2}\).
• If we have to find the resultant gravitational force acting on the particle \(\mathrm{m}\) due to a number of masses \(M_1, M_2 \ldots . M_n\) etc. we use the principle of superposition. Let \(F_1, F_2, \ldots . F_n\) be the individual forces due to \(M_1, M_2, \ldots M_n\), each given by the law of gravitation. From the principle of superposition, each force acts independently and uninfluenced by the other bodies. The resultant force \(F_R\) is then found by vector addition
  \(
  F_R=F_1+F_2+\ldots \ldots+F_n=\sum_{i=1}^n F_i
  \)
  where the symbol ‘ \(\Sigma\) ‘ stands for summation.
• Kepler’s laws of planetary motion state that
  (a) All planets move in elliptical orbits with the Sun at one of the focal points
  (b) The radius vector drawn from the Sun to a planet sweeps out equal areas in equal time intervals. This follows from the fact that the force of gravitation on the planet is central and hence angular momentum is conserved.
  (c) The square of the orbital period of a planet is proportional to the cube of the sem1major axis of the elliptical orbit of the planet
  The period \(T\) and radius \(R\) of the circular orbit of a planet about the Sun are related by
  \(
  T^2=\left(\frac{4 \pi^2}{G M_s}\right) R^3
  \)
  where \(M_5\) is the mass of the Sun. Most planets have nearly circular orbits about the Sun. For elliptical orbits, the above equation is valid if \(R\) is replaced by the sem1-major axis, \(a\)
• The acceleration due to gravity.
  (a) at a height \(h\) above the earth’s surface
  \(
  \begin{aligned}
  & g(h)=\frac{G M_E}{\left(R_E+h\right)^2} \\
  & \approx \frac{G M_E}{R_E^2}\left(1-\frac{2 h}{R_E}\right) \text { for } h \ll R_E \\
  & g(h)=g(0)\left(1-\frac{2 h}{R_E}\right) \text { where } g(0)=\frac{G M_E}{R_E^2}
  \end{aligned}
  \)
  (b) at depth \(d\) below the earth’s surface is
  \(
  g(d)=\frac{G M_E}{R_E^2}\left(1-\frac{d}{R_E}\right)=g(0)\left(1-\frac{d}{R_E}\right)
  \)
• The gravitational force is a conservative force, and therefore a potential energy function can be defined. The gravitational potential energy associated with two particles separated by a distance \(r\) is given by
  \(
  V=-\frac{G m_1 m_2}{r}
  \)
  where \(V\) is taken to be zero at \(r \rightarrow \infty\). The total potential energy for a system of particles is the sum of energies for all pairs of particles, with each pair represented by a term of the form given by the above equation. This prescription follows from the principle of superposition.
• If an isolated system consists of a particle of mass \(m\) moving with a speed \(v\) in the vicinity of a massive body of mass \(M\), the total mechanical energy of the particle is given by
  \(
  E=\frac{1}{2} m v^2-\frac{G M m}{r}
  \)
  That 1s, the total mechanical energy is the sum of the kinetic and potential energies. The total energy is a constant of motion.
• If \(m\) moves in a circular orbit of radius \(a\) about \(M\), where \(M \gg m\), the total energy of the system is
  \(
  E=-\frac{G M m}{2 a}
  \)
  with the choice of the arbitrary constant in the potential energy given in the point 5 ., above. The total energy is negative for any bound system, that is, one in which the orbit is closed, such as an elliptical orbit. The kinetic and potential energies are
  \(
  \begin{aligned}
  & K=\frac{G M m}{2 a} \\
  & V=-\frac{G M m}{a}
  \end{aligned}
  \)
• The escape speed from the surface of the earth is
  \(
  v_e=\sqrt{\frac{2 G M_E}{R_E}}=\sqrt{2 g R_E}
  \)
  and has a value of \(11.2 \mathrm{~km} \mathrm{~s}^{-1}\).
• If a particle is outside a uniform spherical shell or solid sphere with a spherically symmetric internal mass distribution, the sphere attracts the particle as though the mass of the sphere or shell were concentrated at the centre of the sphere.
• If a particle is inside a uniform spherical shell, the gravitational force on the particle is zero. If a particle is inside a homogeneous solid sphere, the force on the particle acts toward the centre of the sphere. This force is exerted by the spherical mass interior to the particle.
• A geostationary (geosynchronous communication) satellite moves in a circular orbit in the equatorial plane at an approximate distance of \(4.22 \times 10^4 \mathrm{~km}\) from the earth’s centre.

\(
\begin{array}{|c|c|c|c|c|}
\hline \text { Physical Quantity } & \text { Symbol } & \text { Dimensions } & \text { Unit } & \text { Remarks } \\
\hline \text { Gravitational Constant } & G & {\left[\mathrm{M}^{-1} \mathrm{~L}^3 \mathrm{~T}^{-2}\right]} & \mathrm{N} \mathrm{m^{2 } \mathrm { Kg } ^ { – 2 }} & 6.67 \times 10^{-1} \\
\hline \begin{array}{l}
\text { Gravitational } \\
\text { Potential Energy }
\end{array} & V(r) & {\left[\mathrm{M} \mathrm{L}^2 \mathrm{~T}^{-2}\right]} & \mathrm{J} & -\frac{G M m}{r} \text { (scalar) }\\
\hline \begin{array}{l}
\text { Gravitational } \\
\text { Potential }
\end{array} & U(r) & {\left[\mathrm{L}^2 \mathrm{~T}^{-2}\right]} & \mathrm{J} \mathrm{kg}^{-1} & -\frac{G M}{r} \text { (scalar) }\\
\hline \begin{array}{l}
\text { Gravitational } \\
\text { Intensity }
\end{array} & \begin{array}{l}
\mathbf{E} \
\text { or } \mathbf{g}
\end{array} & {\left[\mathrm{LT}^{-2}\right]} & \mathrm{m} \mathrm{s}^{-2} & \begin{array}{l}
\frac{G M}{r^2} \hat{\mathbf{r}} \text { (vector) }\\
\end{array} \\
\hline
\end{array}
\)