If amplitude of a particle executing SHM is doubled, which of the following quantities are doubled
(i) Time period
(ii) Maximum velocity
(iii) Maximum acceleration
(iv) Total energy

Options

A. ii & iii
B. i, ii & iii
C. i & iii
D. i, ii, iii & iv

Answer

To solve the problem, we need to analyze how doubling the amplitude of a particle executing Simple Harmonic Motion (SHM) affects various quantities: time period, maximum velocity, maximum acceleration, and total energy.

### Step-by-Step Solution:

1. **Understanding the Initial Conditions**:
- Let the initial amplitude be \( A = a \).
- The new amplitude after doubling is \( A' = 2a \).

2. **Time Period**:
- The time period \( T \) of SHM is given by the formula:
\[ T = 2\pi \sqrt{\frac{m}{k}} \]
- Here, \( T \) does not depend on the amplitude \( A \). Therefore, when the amplitude is doubled, the time period remains unchanged:
\[ T' = T \]
- **Conclusion**: The time period is not doubled.

3. **Maximum Velocity**:
- The maximum velocity \( V_{\text{max}} \) in SHM is given by:
\[ V_{\text{max}} = A \omega \]
- Where \( \omega \) is the angular frequency, defined as:
\[ \omega = \sqrt{\frac{k}{m}} \]
- If we double the amplitude:
\[ V_{\text{max}}' = A' \omega = (2a) \omega = 2(A \omega) = 2V_{\text{max}} \]
- **Conclusion**: The maximum velocity is doubled.

4. **Maximum Acceleration**:
- The maximum acceleration \( a_{\text{max}} \) in SHM is given by:
\[ a_{\text{max}} = A \omega^2 \]
- Doubling the amplitude gives:
\[ a_{\text{max}}' = A' \omega^2 = (2a) \omega^2 = 2(A \omega^2) = 2a_{\text{max}} \]
- **Conclusion**: The maximum acceleration is doubled.

5. **Total Energy**:
- The total energy \( E \) in SHM is given by:
\[ E = \frac{1}{2} k A^2 \]
- If the amplitude is doubled:
\[ E' = \frac{1}{2} k (2a)^2 = \frac{1}{2} k (4a^2) = 4 \left(\frac{1}{2} k a^2\right) = 4E \]
- **Conclusion**: The total energy is quadrupled, not doubled.

### Summary of Results:
- (i) Time period: Not doubled
- (ii) Maximum velocity: Doubled
- (iii) Maximum acceleration: Doubled
- (iv) Total energy: Quadrupled

### Final Answer:
The quantities that are doubled when the amplitude is doubled are:
- Maximum velocity
- Maximum acceleration