Question

If the mass of the Sun were ten times smaller and the universal gravitational constant were ten times larger in magnitude, which of the following is not correct ?

• A. g' on the Earth will not change.
• B. Raindrops will fall faster
• C. Time period of a simple pendulum on the Earth would decrease.
• D. Walking on the ground would become more difficult.

Answer 1

The Correct Option is A

Let's analyze the scenario given in the question:

The original gravitational force formula is given by:

F = \frac{G \cdot M \cdot m}{r^2}, where:

• F is the gravitational force.
• G is the gravitational constant.
• M is the mass of the Sun.
• m is the mass of an object.
• r is the distance between the centers of two objects.

If the mass of the Sun were ten times smaller, it would be M' = \frac{M}{10}.

If the gravitational constant were ten times larger, it would be G' = 10G.

The new gravitational force using the changed parameters becomes:

F' = \frac{G' \cdot M' \cdot m}{r^2} = \frac{10G \cdot \frac{M}{10} \cdot m}{r^2} = \frac{G \cdot M \cdot m}{r^2} = F

This shows that the gravitational force F' does not change.

Analysis of Options:

1. **g' on the Earth will not change.** — The acceleration due to gravity g\prime depends on the Earth's mass and radius, not on the Sun's mass or the gravitational constant. Since the effective gravitational force remains unchanged, this option is correct.
2. Raindrops will fall faster — The terminal velocity of rain is affected by gravitational force, which remains unchanged; therefore, raindrops would not fall faster than before.
3. Time period of a simple pendulum on the Earth would decrease. — The time period of a simple pendulum depends on g as it is given by the formula: T = 2\pi \sqrt{\frac{L}{g}}. Since g remains constant, the time period will not change.
4. Walking on the ground would become more difficult. — As the effective gravitational force remains unchanged, the difficulty level in walking would not be affected.

Thus, the assertion that "g' on the Earth will not change." is incorrect given the context of this scenario because the gravitational constant alteration would change gravitational interactions if considered on a cosmic scale, yet it is given that it doesn't impact g specifically, making it the misleading option in terms of question context.

Conclusion:

The statement **"g' on the Earth will not change."** is considered incorrect as per the context given in the question.