Two simple harmonic motions of angular frequency `100 rad s^(-1)` and `1000 rad s^(-1)` have the same displacement amplitude. The ratio of their maximum accelerations is

Options

A. `1:10`
B. `1:10^(2)`
C. `1:10^(3)`
D. `1:10^(4)`

Answer

To solve the problem, we need to find the ratio of the maximum accelerations of two simple harmonic motions (SHM) with different angular frequencies but the same displacement amplitude.

### Step-by-Step Solution:

1. **Understand the formula for maximum acceleration in SHM**:
The maximum acceleration \( A_{\text{max}} \) of a particle in simple harmonic motion is given by the formula:
\[ A_{\text{max}} = \omega^2 A \]
where \( \omega \) is the angular frequency and \( A \) is the amplitude.

2. **Identify the given values**:
We have two angular frequencies:
- \( \omega_1 = 100 \, \text{rad/s} \)
- \( \omega_2 = 1000 \, \text{rad/s} \)
Both motions have the same amplitude \( A \).

3. **Write the expressions for maximum accelerations**:
For the first motion:
\[ A_{\text{max},1} = \omega_1^2 A = (100)^2 A = 10000 A \]
For the second motion:
\[ A_{\text{max},2} = \omega_2^2 A = (1000)^2 A = 1000000 A \]

4. **Calculate the ratio of maximum accelerations**:
The ratio of the maximum accelerations \( \frac{A_{\text{max},1}}{A_{\text{max},2}} \) is:
\[ \frac{A_{\text{max},1}}{A_{\text{max},2}} = \frac{10000 A}{1000000 A} \]
Since \( A \) is the same for both, it cancels out:
\[ \frac{A_{\text{max},1}}{A_{\text{max},2}} = \frac{10000}{1000000} = \frac{1}{100} \]

5. **Conclusion**:
The ratio of their maximum accelerations is:
\[ \frac{A_{\text{max},1}}{A_{\text{max},2}} = \frac{1}{100} \]

### Final Answer:
The ratio of their maximum accelerations is \( \frac{1}{100} \).

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