Gravitation

The gravitational force at a height from the surface of the planet is given by: where is the gravitational force at the surface of the planet, and is the radius of the planet. For , we get:

The escape velocity is given by: where is the gravitational acceleration and is the radius. The velocity at the equator is , so the escape velocity is .

From Kepler's third law and the law of gravitation, using dimensional analysis and considering the constant G, we find that the correct relation is .

Since the sphere at position D is equidistant from all the spheres in the larger ring, the gravitational forces exerted by all the spheres in the larger ring will cancel out due to symmetry, leading to zero net gravitational force.

The gravitational force on a point due to a ring of mass is given by the formula: where is the gravitational constant, is the total mass of the ring, is the mass experiencing the force, and is the distance from the center of the ring to the mass. To find the position where the forces from both rings are equal, we equate the gravitational forces from both rings.
The forces from the rings at positions A, B, C, and D need to balance.
By symmetry and considering the relative distances, position B is the location where the forces from both rings are equal.
Thus, the correct answer is (b).

The gravitational force from the Earth and the Sun on the rocket will cancel out at a point where the net gravitational force is zero.
The formula for the gravitational force is: where is the gravitational constant, and are the masses, and is the distance between the objects. For zero gravitational force, the force due to Earth’s gravity and the force due to Sun’s gravity must be equal: Simplifying the equation: where is the distance between the Earth and the Sun.
Solving for , we find:
Thus, the correct answer is (c).

The potential energy of the gravitational interaction between a point mass and a uniform rod is calculated by integrating the gravitational potential energy contributions from each infinitesimal segment of the rod. The formula for the potential energy is: Hence, the correct answer is (d).

In this problem, we have two rings of mass, one with a radius and the other with a radius . The gravitational forces from the two rings will cancel each other out at a certain point along the line between the two rings.

To find the point where the net force is zero, we use the principle of superposition, where the force due to each ring will act in opposite directions. The gravitational force due to a ring is proportional to the inverse square of the distance.

At point B, the gravitational forces from both rings are equal in magnitude but opposite in direction, thus cancelling each other out, making the net force zero.

Hence, the correct answer is (b).