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Consider the following statements regarding Kepler's laws of planetary motion and the universal law of gravitation:
- 1. Kepler's second law (law of areal velocities) indicates that the angular momentum of a planet revolving around the Sun remains conserved, because the areal velocity ($dA/dt$) is constant in motion under central forces.
- 2. According to Newton's universal law of gravitation, the gravitational force acting between any two bodies depends on the nature of the medium, and if both bodies are placed underwater, the force of attraction between them becomes zero.
- 3. The 'escape velocity' required to send a body from the surface of the Earth to infinite distance depends on the mass of the body and the angle of projection, and the value of escape velocity is higher for a body of greater mass.
- 4. If the Earth starts rotating at 17 times its present angular velocity, the apparent weight of objects kept at the equator will become zero and the objects there will experience weightlessness.
Which of the above statement(s) is/are completely true?
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Detailed Explanation
Statement 1 is true because Kepler's second law operates under a central force (gravitational force), where angular momentum is always conserved under a central force and the areal velocity ($dA/dt = L/2m$) is constant. Statement 2 is false because Newton's law of gravitation does not depend on the medium; it acts independently of the medium between any two bodies. Statement 3 is false because the formula for escape velocity ($v_e = \sqrt{2gR}$) does not depend on the mass of the body; it depends only on the mass and radius of the planet. Statement 4 is true because the apparent acceleration due to gravity at the equator is $g' = g - R\omega^2$, and when the value of angular velocity $\omega$ becomes equal to or 17 times greater than $\sqrt{g/R}$, the value of $g'$ becomes zero, making objects feel weightless.
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