A body of mass 0.50kg is executing SHM Its period is 0.1 s and amplitude 10cm. When the body is at a distance of 5cm from the mean postion, find i its accelration ii force acting upon it and ii its potential energy.

Answer

To solve the problem step by step, we need to find the acceleration, force, and potential energy of a body executing Simple Harmonic Motion (SHM) given its mass, period, amplitude, and displacement from the mean position.

### Given Data:
- Mass (m) = 0.50 kg
- Period (T) = 0.1 s
- Amplitude (A) = 10 cm = 0.1 m
- Displacement from mean position (y) = 5 cm = 0.05 m

### Step 1: Calculate Angular Frequency (ω)
The angular frequency (ω) is given by the formula:
\[ \omega = \frac{2\pi}{T} \]
Substituting the given period:
\[ \omega = \frac{2\pi}{0.1} = 20\pi \, \text{rad/s} \approx 62.83 \, \text{rad/s} \]

### Step 2: Calculate Acceleration (a)
The acceleration in SHM is given by:
\[ a = -\omega^2 y \]
Substituting the values:
\[ a = - (20\pi)^2 \times 0.05 \]
Calculating:
\[ a = - (400\pi^2) \times 0.05 \approx -62.83 \, \text{m/s}^2 \]
(Note: The negative sign indicates that the acceleration is directed towards the mean position.)

### Step 3: Calculate Force (F)
The force acting on the body can be calculated using Newton's second law:
\[ F = m \cdot a \]
Substituting the values:
\[ F = 0.50 \times (-62.83) \approx -31.415 \, \text{N} \]
(Note: Again, the negative sign indicates that the force is directed towards the mean position.)

### Step 4: Calculate Potential Energy (PE)
The potential energy in SHM is given by:
\[ PE = \frac{1}{2} k y^2 \]
Where \( k = m \omega^2 \). First, we calculate \( k \):
\[ k = m \cdot \omega^2 = 0.50 \cdot (20\pi)^2 = 0.50 \cdot 400\pi^2 \approx 197.88 \, \text{N/m} \]
Now substituting \( k \) and \( y \) into the potential energy formula:
\[ PE = \frac{1}{2} \cdot 197.88 \cdot (0.05)^2 \]
Calculating:
\[ PE = \frac{1}{2} \cdot 197.88 \cdot 0.0025 \approx 0.24735 \, \text{J} \]

### Final Answers:
1. Acceleration (a) ≈ -62.83 m/s²
2. Force (F) ≈ -31.415 N
3. Potential Energy (PE) ≈ 0.24735 J