Question

Find ratio of potential energy of a body at point A to point B for the figure shown. 

• A. \(1:3\)
• B. \(2:3\)
• C. \(3:1\)
• D. \(1:2\)

Hint:

Gravitational potential energy varies as: \[ U \propto -\frac{1}{r} \] If distance triples, potential energy becomes one-third.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
The gravitational potential energy of a body of mass $m$ at a distance $r$ from the center of Earth (mass $M$) is given by Newton's law of universal gravitation.
Step 2: Key Formula or Approach:
The formula for gravitational potential energy is:
\[ U = -\frac{GMm}{r} \]
where $r$ is the radial distance from the center of the Earth.
Step 3: Detailed Explanation:
According to the figure description:
- Point A is at the surface of Earth, so its distance from the center is $r_A = R$.
- Point B is at a distance of $2R$ from the surface (from A), so its total distance from the center is $r_B = R + 2R = 3R$.
Calculate the potential energy at A:
\[ U_A = -\frac{GMm}{R} \]
Calculate the potential energy at B:
\[ U_B = -\frac{GMm}{3R} \]
Now, find the ratio of $U_A$ to $U_B$:
\[ \text{Ratio} = \frac{U_A}{U_B} = \frac{-\frac{GMm}{R}}{-\frac{GMm}{3R}} = \frac{1/R}{1/3R} = \frac{3R}{R} = 3 \]
Step 4: Final Answer:
The ratio is 3.