Pendulums

To solve this problem, we need to determine the speed of the pendulum bob at the lowest point after considering the energy dissipated due to air resistance.

Initially, when the bob is released from the horizontal position, its entire energy is potential energy given by:

where:

• is the mass of the bob (we will see that it cancels out later)
• is the acceleration due to gravity
• is the height

Therefore, the initial energy is:

According to the problem, 10% of the initial energy is dissipated, which implies 90% of the initial energy is converted to kinetic energy at the lowest point. Thus, the energy at the lowest point is:

The kinetic energy of the bob at the lowest point is given by:

Setting the kinetic energy equal to the available energy, we have:

Cancelling from both sides:

Solving for by multiplying both sides by 2:

Taking the square root of both sides, we find:

Therefore, the speed with which the bob arrives at the lowest point is . Hence, the correct answer is:

Applying the conservation of momentum:

Substituting the given values:

The height to which the bob rises can be calculated using:

The time period of a simple pendulum at a height h = R above the earth’s surface is given by:

where the effective acceleration due to gravity at a height is:

Thus, the time period at height is:

Similarly, at height , the effective acceleration due to gravity is:

Hence, the time period is:

The ratio of the time periods is:

This implies:

The time period of a simple pendulum is given by the formula:

where is the length of the pendulum and is the acceleration due to gravity.

Therefore, the correct answer is option (1).