Question

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

• A. planets will have elliptical orbits
• B. Kepler's law of areas does not hold good
• C. 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

Answer 1

Answer: (C)

If the law of gravitation suddenly changes and becomes an inverse cube law, then the law of areas still holds. This is because the force will still be central force, due to which angular momentum will remain constant.
Also, for a planet of mass $m$ revolving around the sun of mass $M$, we can write according to the question,
$F=\frac{GMm}{a^3}=\frac{m{v}^2}{a}$
$[\because$ Centripetal force, $m{\omega }^{2}a=\frac{m{v}^2}{a}]$
where, $a$ is radius of orbiting planet.
Orbital speed, $v=\frac{\sqrt{GM}}{a}\Rightarrow v\propto \frac{1}{a}$
Time period of revolution of a planet,
$T=\frac{2\pi r}{v}$
On putting the value of orbital speed, we get
$T=\frac{2\pi a}{\frac{\sqrt{GM}}{a}}=\frac{2\pi a^2}{\sqrt{GM}}$
$\Rightarrow T^2\propto a^4$
Hence, the orbit of the planet around sun will not be elliptical because for elliptical orbit $T^2\propto a^3$.
Hence, the orbit of the planet around sun will not be elliptical because for elliptical orbit $T^2\propto a^3$.
As force $F=\left(\frac{GM}{a^3}\right)m=g^{\prime }m$, where $g^{\prime }=\frac{GM}{a^3}$.
Since, $g^{\prime }=$ acceleration due to gravity is constant, hence path followed by a projectile will be approximately parabolic $($ as $T\propto a^2)$.