A linear harmonic oscillator of force constant `2 xx 10^(6)N//m` and amplitude `0.01 m` has a total mechanical energy of `160 J`. Its

Options

A. maximum potential energy is 160 J
B. maximum potential energy is 100 J
C. minimum potential energy is zero
D. minimum potential energy is 100 J

Explanation

To solve the problem regarding the linear harmonic oscillator, we will follow these steps:

### Step 1: Understand the Given Data
We have the following information:
- Force constant, \( k = 2 \times 10^6 \, \text{N/m} \)
- Amplitude, \( A = 0.01 \, \text{m} \)
- Total mechanical energy, \( E = 160 \, \text{J} \)

### Step 2: Recall the Formulas
In a harmonic oscillator, the total mechanical energy \( E \) is given by the sum of kinetic energy (KE) and potential energy (PE):
\[ E = KE + PE \]
The potential energy at a displacement \( x \) is given by:
\[ PE = \frac{1}{2} k x^2 \]
The maximum potential energy occurs when \( x = A \) (the amplitude), and the maximum kinetic energy occurs when \( x = 0 \).

### Step 3: Calculate Maximum Potential Energy
At maximum displacement (amplitude), the potential energy is:
\[ PE_{\text{max}} = \frac{1}{2} k A^2 \]
Substituting the values:
\[ PE_{\text{max}} = \frac{1}{2} \times (2 \times 10^6) \times (0.01)^2 \]
Calculating:
\[ PE_{\text{max}} = \frac{1}{2} \times (2 \times 10^6) \times (0.0001) = \frac{1}{2} \times 200 = 100 \, \text{J} \]

### Step 4: Calculate Maximum Kinetic Energy
The maximum kinetic energy occurs when the displacement \( x = 0 \):
\[ KE_{\text{max}} = E - PE_{\text{max}} = 160 \, \text{J} - 100 \, \text{J} = 60 \, \text{J} \]

### Step 5: Determine Minimum Potential Energy
The minimum potential energy occurs when the kinetic energy is at its maximum:
\[ PE_{\text{min}} = E - KE_{\text{max}} = 160 \, \text{J} - 60 \, \text{J} = 100 \, \text{J} \]

### Summary of Results
- Maximum Potential Energy: \( 100 \, \text{J} \)
- Minimum Potential Energy: \( 100 \, \text{J} \)

### Conclusion
Based on the calculations:
- Maximum Potential Energy is \( 100 \, \text{J} \)
- Minimum Potential Energy is \( 100 \, \text{J} \)