Simple Harmonic Motion

The kinetic energy of a particle of mass executing simple harmonic motion (SHM) with angular velocity and amplitude is given by the equation:

Where is the velocity of the particle at a given position. For a particle in SHM, the velocity at a distance from the mean position is: The kinetic energy at distance from the mean position can therefore be written as: We are given that the kinetic energy is . Setting this equal to the expression for kinetic energy, we get: Simplifying the equation: Multiplying both sides by 2: Solving for : Taking the square root: Therefore, the particle is at a distance of from the mean position when its kinetic energy is .

Correct Answer:

Correct Answer: (E)

For simple harmonic motion, the displacement is given by: The velocity is the time derivative of displacement: The maximum velocity occurs when , so: The acceleration is the time derivative of velocity: The maximum acceleration occurs when , so: The ratio of maximum acceleration to maximum velocity is: Thus, the correct ratio is infinity, as tends to infinity when the frequency increases.

The correct option is (D) : infinity

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. Initially, the time period for length is: Now, if the length of the pendulum is increased 4 times, the new length becomes . The new time period is: Thus, the new time period is . The increase in time period is: Thus, the increase in time period is 3 s.

Given parameters:

• Length ± 1 mm
• Time for 100 oscillations ± 1 s

Pendulum equation:

Uncertainty calculations:

Percentage uncertainty:

Thus, the correct option is (D): 2.4.

To solve the problem, we analyze the given displacement equation for simple harmonic motion (SHM):

The general form of the displacement equation for SHM is: where:
and are constants,
is the angular frequency.
The amplitude of the oscillation is given by:
Step 1: Identify , , and From the given equation: we can rewrite it as: Thus:
,
,
.
Step 2: Calculate the amplitude Using the formula for amplitude: Substitute and : Final Answer: The amplitude of oscillation is:

In simple harmonic motion (SHM), the total mechanical energy is conserved and is the sum of the kinetic energy and potential energy .
The total energy is constant and is given by: The total energy in SHM is also related to the amplitude and is given by: where is the mass of the particle and is the angular frequency.
Step 1: Kinetic Energy in SHM The kinetic energy of the particle at a position from the mean position is given by: where is the velocity of the particle at position . The velocity in SHM is related to the displacement by: Thus, the kinetic energy becomes:
Step 2: Kinetic Energy at At , the displacement is times the amplitude. Substituting this into the expression for kinetic energy: Since the total energy , we have: Thus, the kinetic energy at a distance of from the mean position is .

In simple harmonic motion, the kinetic energy and potential energy of the pendulum are equal at the point of maximum amplitude (extreme displacement). At the extreme position, all the energy is potential, and at the equilibrium point, all the energy is kinetic.
Therefore, the ratio of the maximum kinetic energy to the maximum potential energy is 1:1.
Hence, the correct answer is (A).

Step 1: Use the formula for maximum force in simple harmonic motion. The maximum force exerted by the spring is given by the formula: where is the spring constant and is the amplitude.
Step 2: Relate the spring constant to the time period . The time period of a block undergoing simple harmonic motion is related to the spring constant and the mass by the formula: Rearranging for : Substitute the given values and :
Step 3: Calculate the maximum force. Now that we know and , we can calculate the maximum force: Thus, the maximum force exerted by the spring on the block is .

In simple harmonic motion (SHM), the amplitude of the acceleration and displacement are related by the following equations:
- Displacement:
- Acceleration:
From these equations, we can see that the acceleration amplitude is related to the displacement amplitude by:
Therefore, the ratio of the acceleration amplitude to the displacement amplitude is:
Hence, the correct answer is (B).