Relative Motion

The acceleration due to gravity on the surface of a planet is given by: where: - is the gravitational constant, - is the mean density of the planet, - is the radius of the planet. Taking the ratio of gravitational accelerations of the planet ( ) and earth ( ): Given: Since , Thus, the acceleration due to gravity on the planet is .

We apply the concept of the center of mass for a system with a removed section.

Step 1: Define the mass distribution
Let the mass per unit area of the uniform plate be . Then, the mass of the original plate of radius is: The mass of the removed portion (hole) of radius is:

Step 2: Use the concept of the center of mass shift
The center of mass of the remaining portion is found using the formula: Since the center of mass of the original plate is at the origin ( ), and the center of the removed portion is at : Thus, the magnitude of the center of mass shift is:

Step 1: Understanding energy conservation in simple harmonic motion
In simple harmonic motion (SHM), the total mechanical energy is conserved: where: - (spring constant), - cm = m (amplitude of oscillation). Thus, the total energy of the system is: Step 2: Find the kinetic energy at cm
The total mechanical energy in SHM is the sum of kinetic energy and potential energy : Potential energy at displacement cm = m is: Step 3: Compute kinetic energy
Thus, the kinetic energy of the block when it is at a distance of cm from the mean position is J.