Suppose the acceleration due to gravity at the earth's surface is g m/s\(^2\) and at the surface of moon it is g' m/s\(^2\). An M kg passenger goes from the earth to the moon in a spaceship moving with a constant velocity (Neglect all other objects in the sky). Which curve best represents the weight (net gravitational force) as a function of time?
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Remember the "neutral point" in gravitational fields between two massive bodies. The net gravitational force becomes zero there. This is a key feature to look for in such graphs. Also, gravitational force is always non-linear (inverse square law).
Step 1: Understanding the Question:
The question asks to identify the graph that best represents the net gravitational force (weight) experienced by a passenger traveling from Earth to the Moon. Step 3: Detailed Explanation:
1. Starting point (Earth's surface): The passenger's weight is \( Mg \). This is the initial maximum value on the graph.
2. Moving away from Earth: As the spaceship moves away from Earth, the gravitational force due to Earth decreases with the inverse square of the distance (\( F_{Earth} \propto 1/r^2 \)). Simultaneously, the gravitational force due to the Moon (initially very small) starts to increase as the spaceship gets closer to the Moon (\( F_{Moon} \propto 1/(D-r)^2 \)).
3. Neutral Point: There will be a point somewhere between Earth and Moon where the gravitational forces exerted by Earth and Moon are equal and opposite, resulting in a net gravitational force of zero. This point is closer to the Moon because Earth is more massive.
4. Approaching the Moon: After passing the neutral point, the gravitational force due to the Moon becomes dominant and increases as the spaceship approaches the Moon's surface.
5. Moon's surface: The gravitational force on the Moon's surface is \( Mg' \), which is less than \( Mg \) (typically \( g' \approx g/6 \)).
6. Constant Velocity: The mention of "constant velocity" for the spaceship implies that the journey takes a finite time, and the gravitational force profile will be a continuous curve.
Looking at the graphs:
- Curve A and B show a linear or decreasing curve that doesn't reach zero.
- Curve C starts from \( Mg \) and linearly decreases to \( Mg' \), which is incorrect as gravitational force follows an inverse square law.
- Curve D correctly shows:
- Starting at \( Mg \).
- Decreasing non-linearly to zero at some point between Earth and Moon.
- Increasing again non-linearly as it approaches the Moon.
- Ending at a lower value \( Mg' \) on the Moon's surface.
This precisely matches the expected behavior of net gravitational force. Step 4: Final Answer:
Curve D best represents the weight as a function of time.
