Waves And Oscillations

The mechanical energy of a damped oscillator is proportional to the square of its amplitude: Given that the amplitude becomes half of its initial value: The new mechanical energy is: The percentage decrease in energy is:

Step 1: Understanding the SHM Equation

- The given displacement equation follows: - The amplitude of oscillation is .
Step 2: Finding the Distance Traveled from to

- At : - At : - The particle moves from (extreme) → 0 → (extreme) → 0 (mean position). - Total distance traveled = .
Step 3: Finding the Final Position

- At , , which means the particle is at the mean position.
Step 4: Conclusion

Since the total distance traveled is and the final position is the mean position, Option (3) is correct.

Step 1: Understanding the Effect of Cutting the Spring The time period for a mass-spring system is given by: When the spring is cut into three equal parts, each segment will have a new spring constant: Since they are connected in parallel, the effective spring constant becomes: \vspace{0.5cm} Step 2: Finding the New Time Period The new time period when a mass is attached: \vspace{0.5cm} Step 3: Finding the Frequency Frequency is the reciprocal of the time period: Thus, the frequency of oscillation of the combination is: \vspace{0.5cm}

Step 1: Understanding the Concept:
For a particle in Simple Harmonic Motion (SHM), the kinematic quantities are related through the angular frequency .

• Maximum Velocity:
• Maximum Acceleration:

Where is the amplitude.
Step 2: Calculate Angular Frequency ( ):
Given Time Period s.
Step 3: Calculate Maximum Velocity:
We can express in terms of and : Given :
Step 4: Final Answer:
The maximum velocity is .

Step 1: Relate the given force equation to the standard SHM force equation.
The equation for a particle in Simple Harmonic Motion (SHM) is , where is the force constant. The given equation is , which can be rewritten as: By comparing the two equations, we find the force constant: N/m.

Step 2: Calculate the angular frequency .
The angular frequency is related to the force constant and mass by the formula . The mass is given as g kg.

Step 3: State the formula for maximum velocity in SHM.
The maximum velocity ( ) of a particle in SHM is given by the product of its amplitude ( ) and its angular frequency ( ).

Step 4: Calculate the maximum velocity.
We are given the amplitude m. The terms cancel out.

The time period ( ) of a simple pendulum is the time taken for one oscillation. It is related to the length ( ) by the formula:
.
The frequency ( ) is the number of oscillations per unit time, so .
From the formula for T, we have . This shows that or .
Let the initial state be denoted by subscript 1 and the final state by subscript 2.
Initial frequency oscillations per minute.
Final frequency oscillations per minute.
We have the relationship .
.
So, the new length is of the original length . .
The decrease in length is .
The percentage decrease in length is given by:
Percentage decrease = .

Step 1: Understanding the Concept:
In a damped harmonic oscillation, both the amplitude and the total mechanical energy decrease exponentially with time. The key relationship is that the mechanical energy is proportional to the square of the amplitude ( ).
Step 2: Key Formulae:
The amplitude of a damped oscillator at time is given by:
where is the initial amplitude and is a damping constant.
The total mechanical energy of the oscillator at time is:
So, we can write:
where is the initial energy.
Step 3: Detailed Explanation:
Using the Amplitude Information:
We are given that the amplitude becomes half its initial value in 10 s. So, at s, .
Finding the Time for Energy to Halve:
We need to find the time when the energy becomes half its initial value. So, .
Comparing the two expressions:
From equation (1) and (2), we can see that the expressions on the right-hand side are equal. Therefore, their exponents must be equal.
Assuming , we can cancel from both sides.
Step 4: Final Answer:
The time taken for the mechanical energy to become half of its initial value is 5 s. Therefore, option (D) is correct.

1. Total energy in SHM: .
2. At cm, J,
3. Compute : etc.
4. At cm, J.

• At s, .
• At s, .
• Hence, the displacement after one full time interval is zero since both positions correspond to the same equilibrium point.
• Therefore, the net displacement is 0 cm.