Question

If the magnitude of gravitational potential energy of an object of mass $200$ kg at a height of $3.6 \times 10^6$ m from the earth surface is $6 \times 10^6$ J then its value at a height of $5.6 \times 10^6$ m is (Radius of earth is $6.4 \times 10^6$ m)

• A. $5 \times 10^6$ J
• B. $4 \times 10^6$ J
• C. $3 \times 10^6$ J
• D. $2 \times 10^6$ J
• E. $10^6$ J

Hint:

Gravitational potential energy varies inversely with distance: - $U \propto \frac{1}{r}$ - Always take distance from earth’s center $(R+h)$

Answer 1

The Correct Option is A

Concept: Gravitational potential energy: \[ U = -\frac{GMm}{r} \] Magnitude: \[ |U| = \frac{GMm}{r} \Rightarrow |U| \propto \frac{1}{r} \]

Step 1: Find distances from earth’s center.
\[ r_1 = R + h_1 = 6.4\times10^6 + 3.6\times10^6 = 10\times10^6\ \text{m} \] \[ r_2 = R + h_2 = 6.4\times10^6 + 5.6\times10^6 = 12\times10^6\ \text{m} \]

Step 2: Use proportionality.
\[ \frac{U_2}{U_1} = \frac{r_1}{r_2} = \frac{10}{12} = \frac{5}{6} \]

Step 3: Calculate new potential energy.
\[ U_2 = \frac{5}{6} \times 6\times10^6 = 5\times10^6\ \text{J} \]