Gravitational Force

Step 1: Understanding weightlessness in a satellite.
When a person is inside a satellite orbiting the Earth, both the satellite and the person inside it are in free fall. The satellite is constantly falling towards Earth, but it has a high horizontal velocity, so it keeps missing Earth and stays in orbit. The person inside the satellite, being inside the same frame of reference, experiences no contact force from the satellite’s surface and hence feels weightless.
Step 2: Analyze each statement.
(1) The Earth does not attract the object in a satellite: This is false. The Earth does attract the satellite and the person inside, but because both are in free fall, there is no sensation of weight.
(2) The reaction on the person balances the gravitational force: This is incorrect. The reaction force (the normal force) does not balance the gravitational force in the satellite, because the person inside is in free fall, and thus, there is no normal force acting on them.
(3) A person sitting in the satellite is not accelerated: This is misleading. The person inside the satellite is indeed accelerating with the satellite due to gravity, but they do not feel this acceleration because they are in free fall along with the satellite. This is the reason why they feel weightless.
(4) The normal force (reaction) is zero: This is true. In a satellite orbiting Earth, the person inside experiences zero normal force from the surface of the satellite because both the person and the satellite are in free fall. Since there is no normal force acting on the person, they feel weightless.
Step 3: Conclusion.
The reason a person feels weightless in a satellite is because there is no normal force acting on them. Therefore, the correct answer is (4) the normal force (reaction) is zero.

Step 1: Understand the scenario.
In this case, the particle is released inside a tunnel that runs through the Earth and is assumed to execute simple harmonic motion (SHM) under the influence of gravity. This SHM occurs because the force acting on the particle is proportional to its displacement from the center of the Earth, as the gravitational force inside the Earth follows the form of SHM.
Step 2: Force inside the Earth.
The gravitational force at a distance from the center of the Earth (where , being the Earth's radius) is given by: Where:
is the force,
is the gravitational constant,
is the mass of the Earth,
is the mass of the particle,
is the distance from the center of the Earth, and
is the Earth's radius.
This equation indicates that the force is proportional to , which is characteristic of simple harmonic motion.
Step 3: Equation of motion.
For SHM, the general equation is: where is the force constant. From the above equation for force, we can identify .
Step 4: Time period of SHM.
The time period for SHM is given by the formula: Substituting into this equation: Thus, the time period of SHM is , which corresponds to option (3).