Question

The gravitational potential due to earth at infinite distance from it is zero. Let the gravitational potential at a point $\text{P}$ be $-5{\text{Jkg}}^{-1}$. Suppose, we arbitrarily assume the gravitational potential at infinity to be $+10{\text{Jkg}}^{-1}$ , then the gravitational potential at $\text{P}$ will be

• A. $-5{\text{J kg}}^{-1}$
• B. $+5{\text{Jkg}}^{-1}$
• C. $-15{\text{Jkg}}^{-1}$
• D. $+15{\text{Jkg}}^{-1}$

Answer 1

The correct option is B $+5{\text{Jkg}}^{-1}$

$\text{Case I}:$
Gravitational Potential due to Earth at infinity, $V_{\infty }=0$,

Gravitational Potential due to Earth at point $\text{P}$, $V_{\text{P}}=5{\text{Jkg}}^{-1}$,

So, the change in potential,
$\Delta V=V_{\text{P}}-V_{\infty }=-5-0=-5{\text{Jkg}}^{-1}$

Simlarly, for $\text{Case II}:$

Gravitational Potential due to Earth at infinity, $V_{\infty }=10{\text{Jkg}}^{-1}$,

The change in gravitational potential doesn’t depend upon the reference points, so the change will remain same.

So, the change in potential, $\Delta V=-5$

$\Rightarrow V_{\text{P}}-V_{\infty }=-5$

$\Rightarrow V_{\text{P}}-10=-5$

$\Rightarrow V_{\text{P}}=-5+10$

$\therefore V_{\text{P}}=+5{\text{Jkg}}^{-1}$

Hence, option (b) is the correct answer.

$\text{Alternative Solution:}$

According to the problem as the potential at infinity increases by $+10{\text{J kg}}^{-1}$.

The potential will increase by the same amount everywhere (since, potential gradient will remain constant).

Hence, new potential point at point $\text{P}$$=10-5=+5{\text{J kg}}^{-1}$

Why this question: To familiarize students with the concept of gravitational potential and that it doesn’t depend upon the reference chosen.