A particle of mass 1 kg is moving in SHM with path length 0.01 m and a frequency of 50 Hz. The maximum force in newton, acting on the particle is

Options

A. `150pi^2`
B. `200pi^2`
C. `100pi^2`
D. `50pi^2`

Answer

To solve the problem step by step, we will calculate the maximum force acting on a particle moving in Simple Harmonic Motion (SHM) given its mass, path length, and frequency.

### Step 1: Identify the given values
- Mass of the particle, \( m = 1 \, \text{kg} \)
- Path length (which is the amplitude, \( A \)), \( A = 0.01 \, \text{m} \)
- Frequency, \( f = 50 \, \text{Hz} \)

### Step 2: Calculate the angular frequency \( \omega \)
The angular frequency \( \omega \) is given by the formula:
\[ \omega = 2\pi f \]
Substituting the value of frequency:
\[ \omega = 2\pi \times 50 = 100\pi \, \text{rad/s} \]

### Step 3: Calculate the maximum acceleration \( a_{\text{max}} \)
The maximum acceleration in SHM is given by:
\[ a_{\text{max}} = \omega^2 A \]
Substituting the values of \( \omega \) and \( A \):
\[ a_{\text{max}} = (100\pi)^2 \times 0.01 \]
Calculating \( (100\pi)^2 \):
\[ (100\pi)^2 = 10000\pi^2 \]
Now substituting back:
\[ a_{\text{max}} = 10000\pi^2 \times 0.01 = 100\pi^2 \, \text{m/s}^2 \]

### Step 4: Calculate the maximum force \( F_{\text{max}} \)
The maximum force acting on the particle is given by Newton's second law:
\[ F_{\text{max}} = m \cdot a_{\text{max}} \]
Substituting the values of \( m \) and \( a_{\text{max}} \):
\[ F_{\text{max}} = 1 \cdot 100\pi^2 = 100\pi^2 \, \text{N} \]

### Step 5: Final result
Thus, the maximum force acting on the particle is:
\[ F_{\text{max}} = 100\pi^2 \, \text{N} \]