If a particle executes SHM of time period 4 s and amplitude 2 cm, find its maximum velocity and that at half its full displacement also find the acceleration at the turning points and when the displacement is 0.75 cm

Answer

To solve the problem step by step, we will follow the guidelines for simple harmonic motion (SHM) and apply the relevant formulas.

### Given Data:
- Time period (T) = 4 seconds
- Amplitude (A) = 2 cm

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

**Calculation:**
1. Calculate \(\omega\):
\[ \omega = \frac{2\pi}{4} = \frac{\pi}{2} \text{ rad/s} \approx 1.57 \text{ rad/s} \]
2. Now, substitute \(\omega\) and \(A\) into the Vmax formula:
\[ V_{max} = 1.57 \cdot 2 = 3.14 \text{ cm/s} \]

### Step 2: Find the Velocity at Half the Full Displacement (y = 1 cm)
The formula for velocity at a displacement \(y\) is:
\[ V = \omega \sqrt{A^2 - y^2} \]

**Calculation:**
1. Substitute \(A = 2\) cm and \(y = 1\) cm:
\[ V = 1.57 \sqrt{2^2 - 1^2} = 1.57 \sqrt{4 - 1} = 1.57 \sqrt{3} \]
2. Calculate \(V\):
\[ V \approx 1.57 \cdot 1.732 \approx 2.719 \text{ cm/s} \]

### Step 3: Find the Maximum Acceleration (Amax)
The formula for maximum acceleration in SHM is:
\[ A_{max} = \omega^2 \cdot A \]

**Calculation:**
1. Calculate \(A_{max}\):
\[ A_{max} = (1.57)^2 \cdot 2 = 2.4649 \cdot 2 \approx 4.93 \text{ cm/s}^2 \]

### Step 4: Find the Acceleration at Displacement y = 0.75 cm
The formula for acceleration at a displacement \(y\) is:
\[ A = -\omega^2 \cdot y \]

**Calculation:**
1. Substitute \(y = 0.75\) cm:
\[ A = - (1.57)^2 \cdot 0.75 = -2.4649 \cdot 0.75 \approx -1.8487 \text{ cm/s}^2 \]

### Summary of Results:
- Maximum Velocity \(V_{max} = 3.14 \text{ cm/s}\)
- Velocity at \(y = 1 \text{ cm} = 2.719 \text{ cm/s}\)
- Maximum Acceleration \(A_{max} = 4.93 \text{ cm/s}^2\)
- Acceleration at \(y = 0.75 \text{ cm} \approx -1.85 \text{ cm/s}^2\)