Ncert Exercises & Solutions

If the law of gravitation, instead of being inverse square law, becomes an inverse cube law:

(a)	planets will not have elliptic orbits.
(b)	circular orbits of planets are not possible.
(c)	the projectile motion of a stone thrown by hand on the surface of the earth will be approximately parabolic.
(d)	there will be no gravitational force inside a spherical shell of uniform density.

Choose the correct alternatives:
1. (a), (d)
2. (a), (c)
3. (c), (d)
4. (b), (d)

Hint: Apply Kepler's laws.

Step 1: Find the relation between the time period and the radius of the orbit of the planet.

If the law of gravitation becomes an inverse cube law, then we can write, for a planet of mass m revolving around the sun of mass M,

F = \frac{G M m}{a^{3}} = \frac{m v^{2}}{a}

(where a is the radius of orbiting planet)

\Rightarrow

v=orbital speed=

\frac{\sqrt{G M}}{a} \Rightarrow v \propto \frac{1}{a}

The time period of revolution of a planet T=

\frac{2 πa}{v} = \frac{2 πa}{\frac{\sqrt{G M}}{a}} = \frac{2 {πa}^{2}}{\sqrt{G M}}

\Rightarrow

T^{2} \propto a^{4}

Hence, the orbit will not be elliptical. [for elliptical orbit,

T^{2} \propto a^{3}

]

Step 2: Find the acceleration due to gravity to find the path of the projectile.

As the force,

F = (\frac{G M}{a^{3}}) m = g' m

where,

g' = \frac{G M}{a^{3}}

As g', acceleration due to gravity is constant, hence the path followed by a projectile will be approximately parabolic.

(a s T \propto a^{2})