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

Answer

To solve the problem of finding the maximum potential energy and maximum kinetic energy of a linear harmonic oscillator, we can follow these steps:

### Step 1: Understand the Total Mechanical Energy
The total mechanical energy (E) in a simple harmonic oscillator is given as:
\[ E = \text{Maximum Potential Energy} + \text{Maximum Kinetic Energy} \]
In this case, the total mechanical energy is provided as \( E = 160 \, \text{J} \).

### Step 2: Determine Maximum Potential Energy
At the extreme positions of the oscillator (maximum displacement), the kinetic energy is zero, and all the energy is potential. Thus, the maximum potential energy (U_max) is equal to the total mechanical energy:
\[ U_{\text{max}} = E = 160 \, \text{J} \]

### Step 3: Determine Maximum Kinetic Energy
At the mean position, the potential energy is zero, and all the energy is kinetic. Thus, the maximum kinetic energy (K_max) can also be calculated using the total mechanical energy:
\[ K_{\text{max}} = E - U_{\text{max}} \]
Since \( U_{\text{max}} = 160 \, \text{J} \):
\[ K_{\text{max}} = 160 \, \text{J} - 160 \, \text{J} = 0 \, \text{J} \]

However, we can also calculate the maximum kinetic energy using the formula:
\[ K_{\text{max}} = \frac{1}{2} k A^2 \]
Where:
- \( k = 2 \times 10^6 \, \text{N/m} \) (force constant)
- \( A = 0.01 \, \text{m} \) (amplitude)

### Step 4: Calculate Maximum Kinetic Energy
Substituting the values into the formula:
\[ K_{\text{max}} = \frac{1}{2} \times (2 \times 10^6) \times (0.01)^2 \]
\[ K_{\text{max}} = \frac{1}{2} \times (2 \times 10^6) \times (1 \times 10^{-4}) \]
\[ K_{\text{max}} = \frac{1}{2} \times 200 \]
\[ K_{\text{max}} = 100 \, \text{J} \]

### Final Results
- Maximum Potential Energy \( U_{\text{max}} = 160 \, \text{J} \)
- Maximum Kinetic Energy \( K_{\text{max}} = 100 \, \text{J} \)

### Summary
- Maximum Potential Energy: **160 J**
- Maximum Kinetic Energy: **100 J**