Question

13. If the gravitational force had varied as r 2 instead
of r-2. the potential energy of a particle at a
distance from centre of earth would be
proportional to
(1)r-1
(2) r-5/2
(3)r-3/2
(4)r-2​

Answer 1

Answer:

None of the given option is correct. The correct answer is $\rm r^3.$

Explanation:

The Gravitational force $\rm F_g$ and the potential energy $\rm U$ of a particle are related as

$\rm F_g = -\dfrac{dU}{dr}\\\therefore U=-\int\limits^r_0F_g\ dr'.$

The gravitational force on the particle at r distance from the surface of the Earth is given as

$\rm F_g = \dfrac{GMm}{r^2}.$

• G = Universal Gravtitational constant.
• M = mass of the Earth.
• m = mass of the particle.

If the gravitational force had varied as $\rm r^2$ instead  of $\rm r^{-2}$, then, $\rm F_g = GMmr^2.$

In that case,

$\rm U=-\int\limits^r_0(GMmr'^2)\ dr'\\=-GMm\int\limits^r_0r'^2\ dr'\\=-GMm\left ( \dfrac{r'^3}{3}\right )\limits^r_0\\=-GMm\left ( \dfrac{r^3}{3}-0\right )\\=-GMm\left ( \dfrac{r^3}{3}\right ).\\\Rightarrow U\propto r^3.$

Answer 2

Answer:

I hope this is the correct answer ☺️