Two simple harmonic motion of angular frequency `100and 1000 rads^(-1)` have the same displacement amplitude The ratio of their maximum acceleration is

Options

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

Explanation

To find the ratio of the maximum accelerations of two simple harmonic motions (SHM) with angular frequencies of \( \omega_1 = 100 \, \text{rad/s} \) and \( \omega_2 = 1000 \, \text{rad/s} \), we can follow these steps:

### Step 1: Understand the formula for maximum acceleration in SHM
The maximum acceleration \( A_{\text{max}} \) 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 displacement amplitude.

### Step 2: Write the expressions for maximum acceleration for both SHMs
For the first SHM with \( \omega_1 = 100 \, \text{rad/s} \):
\[ A_{1} = -\omega_1^2 A = -100^2 A = -10000 A \]

For the second SHM with \( \omega_2 = 1000 \, \text{rad/s} \):
\[ A_{2} = -\omega_2^2 A = -1000^2 A = -1000000 A \]

### Step 3: Calculate the ratio of maximum accelerations
Now, we can find the ratio of the maximum accelerations \( \frac{A_1}{A_2} \):
\[ \frac{A_1}{A_2} = \frac{-10000 A}{-1000000 A} \]
The negative signs and the amplitude \( A \) cancel out:
\[ \frac{A_1}{A_2} = \frac{10000}{1000000} = \frac{1}{100} \]

### Step 4: Simplify the ratio
This can be expressed as:
\[ \frac{A_1}{A_2} = \frac{1}{10^2} \]

### Final Answer
Thus, the ratio of their maximum accelerations is:
\[ \frac{A_1}{A_2} = \frac{1}{100} \]