Simple Harmonic Motion

Step 1: Identify the angular frequency . The angular frequency from the equation is .
Step 2: Calculate the time period . The time period of simple harmonic motion is given by .

We are given the following data:
- The time period of the spring without the additional weight is ,
- The time period of the spring with the additional weight is .
The time period of a mass-spring system is given by the formula: where is the time period, is the mass, and is the spring constant.
Step 1: Find the ratio of the time periods. Using the relationship between the initial and final time periods: Squaring both sides: Substitute the values: Thus, the mass after adding the weight is times the original mass. The additional mass is: Step 2: Find the extension. The extension of the spring due to the additional weight is proportional to the weight .
Using the force extension relationship: The extension is: Thus, the extension due to the additional weight is 3 cm.
Conclusion: The extension due to the additional weight is 3 cm, so the correct answer is (2) 3 cm.

For simple harmonic motion, the maximum acceleration is given by: Where: - is the amplitude, - is the angular frequency.
Thus: Final Answer:

Step 1: The amplitude of the damped oscillator is given by the exponential term in the displacement equation: where is the initial amplitude.

Step 2: We are asked to find the time when the amplitude becomes times its initial amplitude. This means: Substitute the expression for :

Step 3: Cancel from both sides: Taking the natural logarithm of both sides: Thus, the time required for the amplitude to become times its initial amplitude is 6 s.