Maximum kinetic energy of a particle of mass 1 kg in SHM is 8 J. Time period of SHM is 4 s. Maximum potential energy during the motion is 10 J. Then

Options

A. amplitude of oscillation is approximately 2.53 m
B. minimum potential energy of the particle is 2 J
C. maximum acceleration of the particle is approximately `6.3 ms^(-2)`
D. minimum kinetic energy of the particle is 2 J

Explanation

To solve the problem step by step, we will analyze the given data and use the relevant formulas for Simple Harmonic Motion (SHM).

### Given Data:
- Mass (m) = 1 kg
- Maximum Kinetic Energy (KE_max) = 8 J
- Time Period (T) = 4 s
- Maximum Potential Energy (PE_max) = 10 J

### Step 1: Calculate the Spring Constant (k)
The total mechanical energy (E) in SHM is the sum of maximum kinetic energy and maximum potential energy:
\[ E = KE_{max} + PE_{max} \]
\[ E = 8 J + 10 J = 18 J \]

The total energy in SHM is also given by:
\[ E = \frac{1}{2} k A^2 \]
Where A is the amplitude.

Setting the two expressions for energy equal:
\[ 18 = \frac{1}{2} k A^2 \]
\[ k A^2 = 36 \quad (1) \]

### Step 2: Relate Time Period to Spring Constant
The time period (T) of SHM is given by:
\[ T = 2\pi \sqrt{\frac{m}{k}} \]
Rearranging gives:
\[ k = \frac{4\pi^2 m}{T^2} \]

Substituting the values of m and T:
\[ k = \frac{4\pi^2 \cdot 1}{4^2} = \frac{4\pi^2}{16} = \frac{\pi^2}{4} \quad (2) \]

### Step 3: Substitute k into Equation (1)
Now we substitute the value of k from equation (2) into equation (1):
\[ \frac{\pi^2}{4} A^2 = 36 \]
\[ A^2 = \frac{36 \cdot 4}{\pi^2} \]
\[ A^2 = \frac{144}{\pi^2} \]
\[ A = \frac{12}{\pi} \quad (3) \]

### Step 4: Calculate Maximum Acceleration
The maximum acceleration (a_max) in SHM is given by:
\[ a_{max} = \omega^2 A \]
Where \(\omega = \frac{2\pi}{T}\).

Calculating \(\omega\):
\[ \omega = \frac{2\pi}{4} = \frac{\pi}{2} \]

Now substituting \(\omega\) and \(A\) into the equation for maximum acceleration:
\[ a_{max} = \left(\frac{\pi}{2}\right)^2 \cdot \frac{12}{\pi} \]
\[ a_{max} = \frac{\pi^2}{4} \cdot \frac{12}{\pi} = 3\pi \]

### Final Results
- Maximum Acceleration \(a_{max} = 3\pi \, \text{m/s}^2\)
- Spring Constant \(k = \frac{\pi^2}{4} \, \text{N/m}\)