Question

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

• A. planets will not have elliptic orbits.
• B. projectile motion of a stone thrown by hand on the surface of the earth will be approximately parabolic.
• C. circular orbits of planets is not possible.
• D. there will be no gravitational force inside a spherical shell of uniform density

Answer 1

Answer: (A), (B), (C)

circular orbits of planets is not possible.

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

$F=\frac{GMm}{R^3}=\frac{m{v}^2}{R}$

(Where $R$ is the radius of orbiting planet)

Orbital speed

$V=\frac{\sqrt{GM}}{R}$

$V\propto \frac{1}{R}$

$\left(a\right)$ Time period of revolution of a planet

$T=\frac{2\pi R}{V}$

$=\frac{2\pi R}{\frac{\sqrt{GM}}{R}}$

$=\frac{2\pi R^2}{\sqrt{GM}}$



$T^2\propto R^4$

For elliptical orbit $T^2\propto R^3$ where $R$ is semi-major axis,

Hence, orbit will not be elliptical.

$\left(b\right)$ For Circular orbit $T^2\propto R^3$

So,the circular orbits of the planets are not possible according to new law of gravitation.

$\left(c\right)$ As force $F=\left(\frac{Gmm}{R^3}\right)$

Acceleration due to gravity $g^{\prime }=\frac{F}{m}=\frac{GM}{R^3}$

As $g^{{}^{\prime }}$ is constant and towards the centre of earth, hence path followed by a projectile will be approximately parabolic.

$\left(d\right)$ The gravitational field $g^{{}^{\prime }}$ inside a spherical shell of uniform density $\left(\rho \right)$ at a distance r due $M^{{}^{\prime }}$ $\Rightarrow g^{{}^{\prime }}=\frac{G{M}^{{}^{\prime }}}{r^3}$

So, gravitational force at a distance $r$
$F=m{g}^{\prime }$

So, gravitational force inside a spherical shell of uniform density will have some value. So, only option $\left(d\right)$ is incorrect.

Final Answer: $\left(a\right),\left(b\right),\left(c\right)$