A particle of mass 10grams is executing simple harmonic motion with an amplitude of 0.5 m and periodic time of )`(pi //5)`seconds . The maximum value of the force acting on the particle

Options

A. 25 N
B. 5 N
C. 2.5 N
D. 0.5 N

Answer

To find the maximum value of the force acting on a particle executing simple harmonic motion (SHM), we can follow these steps:

### Step 1: Convert Mass to Kilograms
The mass of the particle is given as 10 grams. We need to convert this to kilograms since the standard unit of mass in physics is kilograms.

\[ \text{Mass (m)} = 10 \text{ grams} = 10 \times 10^{-3} \text{ kg} = 0.01 \text{ kg} \]

### Step 2: Calculate Angular Frequency (ω)
The angular frequency (ω) can be calculated using the formula:

\[ \omega = \frac{2\pi}{T} \]

where \( T \) is the periodic time. Given \( T = \frac{\pi}{5} \) seconds, we can substitute this value:

\[ \omega = \frac{2\pi}{\frac{\pi}{5}} = 2\pi \times \frac{5}{\pi} = 10 \text{ rad/s} \]

### Step 3: Calculate Maximum Acceleration (A_max)
The maximum acceleration in simple harmonic motion is given by the formula:

\[ A_{\text{max}} = \omega^2 A \]

where \( A \) is the amplitude. Given \( A = 0.5 \) m, we can substitute the values:

\[ A_{\text{max}} = (10)^2 \times 0.5 = 100 \times 0.5 = 50 \text{ m/s}^2 \]

### Step 4: Calculate Maximum Force (F_max)
The maximum force acting on the particle can be calculated using Newton's second law:

\[ F_{\text{max}} = m \cdot A_{\text{max}} \]

Substituting the values we have:

\[ F_{\text{max}} = 0.01 \text{ kg} \times 50 \text{ m/s}^2 = 0.5 \text{ N} \]

### Final Answer
The maximum value of the force acting on the particle is:

\[ F_{\text{max}} = 0.5 \text{ N} \]

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