A `0.5 kg` body performs simple harmonic motion with a frequency of `2 Hz` and an amplitude of `8 mm`. Find the maximum velocity of the body, its maximum acceleration and the maximum restoring force to which the body is subjected.

Explanation

To solve the problem step by step, we will find the maximum velocity, maximum acceleration, and maximum restoring force for a body performing simple harmonic motion (SHM) given its mass, frequency, and amplitude.

### Given Data:
- Mass (m) = 0.5 kg
- Frequency (f) = 2 Hz
- Amplitude (A) = 8 mm = 8 × 10^-3 m

### Step 1: Calculate Maximum Velocity (Vmax)
The formula for maximum velocity in SHM is given by:
\[ V_{max} = \omega A \]
Where:
- \( \omega \) (angular frequency) can be calculated using the formula:
\[ \omega = 2 \pi f \]

**Calculating \( \omega \):**
\[ \omega = 2 \pi (2) = 4 \pi \, \text{rad/s} \]

**Now substituting \( \omega \) and \( A \) into the maximum velocity formula:**
\[ V_{max} = (4 \pi)(8 \times 10^{-3}) \]

**Calculating:**
\[ V_{max} = 32 \pi \times 10^{-3} \]
\[ V_{max} \approx 0.101 \, \text{m/s} \]

### Step 2: Calculate Maximum Acceleration (Amax)
The formula for maximum acceleration in SHM is given by:
\[ A_{max} = \omega^2 A \]

**Substituting \( \omega \) and \( A \):**
\[ A_{max} = (4 \pi)^2 (8 \times 10^{-3}) \]

**Calculating:**
\[ A_{max} = 16 \pi^2 (8 \times 10^{-3}) \]
\[ A_{max} \approx 1.264 \, \text{m/s}^2 \]

### Step 3: Calculate Maximum Restoring Force (Fmax)
The maximum restoring force can be calculated using Newton's second law:
\[ F_{max} = m A_{max} \]

**Substituting the values:**
\[ F_{max} = (0.5)(1.264) \]

**Calculating:**
\[ F_{max} \approx 0.632 \, \text{N} \]

### Final Answers:
- Maximum Velocity (Vmax) ≈ 0.101 m/s
- Maximum Acceleration (Amax) ≈ 1.264 m/s²
- Maximum Restoring Force (Fmax) ≈ 0.632 N

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