If amplitude of a particle in S.H.M. is doubled, which of the following quantities will be doubled

Options

A. time period
B. total energy
C. maximum acceleration
D. maximum velocity

Explanation

To solve the problem, we need to analyze how the doubling of amplitude in Simple Harmonic Motion (S.H.M.) affects various quantities associated with the motion.

### Step-by-Step Solution:

1. **Understanding the Amplitude in S.H.M.**:
- In S.H.M., the amplitude (A) is the maximum displacement from the mean position.

2. **Time Period (T)**:
- The time period \( T \) of S.H.M. is given by the formula:
\[ T = 2\pi \sqrt{\frac{m}{k}} \]
- Here, \( m \) is the mass and \( k \) is the spring constant.
- Notice that the amplitude \( A \) does not appear in this formula. Therefore, if the amplitude is doubled, the time period remains unchanged.
- **Conclusion**: Time period does not change.

3. **Total Energy (E)**:
- The total energy \( E \) in S.H.M. is given by:
\[ E = \frac{1}{2} m \omega^2 A^2 \]
- Where \( \omega \) is the angular frequency. If we double the amplitude \( A \), the new energy becomes:
\[ E' = \frac{1}{2} m \omega^2 (2A)^2 = \frac{1}{2} m \omega^2 (4A^2) = 4E \]
- **Conclusion**: Total energy increases by a factor of 4 when amplitude is doubled.

4. **Maximum Acceleration (a_max)**:
- The maximum acceleration \( a_{max} \) is given by:
\[ a_{max} = -\omega^2 A \]
- If we double the amplitude \( A \), the new maximum acceleration becomes:
\[ a_{max}' = -\omega^2 (2A) = 2(-\omega^2 A) = 2a_{max} \]
- **Conclusion**: Maximum acceleration doubles when amplitude is doubled.

5. **Maximum Velocity (v_max)**:
- The maximum velocity \( v_{max} \) is given by:
\[ v_{max} = \omega A \]
- If we double the amplitude \( A \), the new maximum velocity becomes:
\[ v_{max}' = \omega (2A) = 2(\omega A) = 2v_{max} \]
- **Conclusion**: Maximum velocity also doubles when amplitude is doubled.

### Final Conclusions:
- The quantities that will be doubled when the amplitude is doubled are:
- Maximum Acceleration
- Maximum Velocity