A block is kept on a horizontal table. The stable is undergoing simple harmonic motion of frequency `3Hz` in a horizontal plane. The coefficient of static friciton between block and the table surface is `0.72.` find the maximum amplitude of the table at which the block does not slip on the surface.

Options

A. 3
B. 1
C. 2
D. 7

Explanation

To solve the problem step by step, we need to find the maximum amplitude of the table's simple harmonic motion such that the block does not slip off. Here’s how we can approach the solution:

### Step 1: Identify the given values
- Frequency of motion, \( f = 3 \, \text{Hz} \)
- Coefficient of static friction, \( \mu_s = 0.72 \)
- Acceleration due to gravity, \( g = 10 \, \text{m/s}^2 \)

### Step 2: Calculate the angular frequency
The angular frequency \( \omega \) can be calculated using the formula:
\[ \omega = 2\pi f \]
Substituting the value of \( f \):
\[ \omega = 2\pi \times 3 = 6\pi \, \text{rad/s} \]

### Step 3: Relate maximum acceleration to static friction
The maximum acceleration \( a_{\text{max}} \) that can be sustained without slipping is given by:
\[ a_{\text{max}} = \mu_s g \]
Substituting the values:
\[ a_{\text{max}} = 0.72 \times 10 = 7.2 \, \text{m/s}^2 \]

### Step 4: Relate maximum acceleration to amplitude
The maximum acceleration in simple harmonic motion is also given by:
\[ a_{\text{max}} = \omega^2 A \]
Where \( A \) is the amplitude. Rearranging this equation gives:
\[ A = \frac{a_{\text{max}}}{\omega^2} \]

### Step 5: Substitute the values to find amplitude
Now substituting \( a_{\text{max}} \) and \( \omega \):
\[ A = \frac{7.2}{(6\pi)^2} \]
Calculating \( (6\pi)^2 \):
\[ (6\pi)^2 = 36\pi^2 \approx 36 \times 9.87 \approx 355.32 \]
Now substituting back:
\[ A = \frac{7.2}{355.32} \approx 0.0203 \, \text{m} \]

### Step 6: Convert to centimeters
To convert meters to centimeters:
\[ A \approx 0.0203 \, \text{m} \times 100 = 2.03 \, \text{cm} \]

### Final Answer
The maximum amplitude of the table at which the block does not slip on the surface is approximately:
\[ \boxed{2 \, \text{cm}} \]