Question

The magnitude of gravitational potential energy of a body at a distance ' \( R \) ' from the centre of the earth is ' \( E \) '. Its weight at a distance ' \( 1.5 R \) ' from the centre of the earth is

• A. \( \frac{2E}{9R} \)
• B. \( \frac{4E}{5R} \)
• C. \( \frac{4E}{9R} \)
• D. \( \frac{2E}{7R} \)

Hint:

Potential Energy $U \propto \frac{1}{r}$ and Weight $W \propto \frac{1}{r^2}$. Thus, $W = \frac{|U|}{r}$.

Answer 1

The Correct Option is C

Step 1: Potential Energy Relation
Magnitude of Potential Energy at distance $R$: $E = \frac{GMm}{R} \implies GMm = ER$.
Step 2: Weight Formula
Weight $W = F_g = \frac{GMm}{r^2}$.
At distance $r = 1.5R = \frac{3}{2}R$, the weight is:
$W = \frac{GMm}{(\frac{3}{2}R)^2} = \frac{GMm}{\frac{9}{4}R^2} = \frac{4GMm}{9R^2}$.
Step 3: Substitution
Substitute $GMm = ER$:
$W = \frac{4(ER)}{9R^2} = \frac{4E}{9R}$.
Final Answer: (C)