An oscillating mass spring system has mechanical energy 1 joule, when it has an amplitude 0.1 m and maximum speed of `1ms^(-1)`. Then the force constant of the spring is

Options

A. 100 `Nm^(-1)`
B. 200 `Nm^(-1)`
C. 300 `Nm^(-1)`
D. 50 `Nm^(-1)`

Explanation

To find the force constant \( k \) of the spring in the oscillating mass-spring system, we can follow these steps:

### Step 1: Understand the given information
We are given:
- Mechanical energy \( E = 1 \, \text{J} \)
- Amplitude \( A = 0.1 \, \text{m} \)
- Maximum speed \( V_{\text{max}} = 1 \, \text{m/s} \)

### Step 2: Use the formula for mechanical energy in a spring-mass system
The mechanical energy \( E \) in a mass-spring system is given by the formula:
\[ E = \frac{1}{2} k A^2 \]
where \( k \) is the force constant and \( A \) is the amplitude.

### Step 3: Substitute the known values into the energy formula
We can rearrange the formula to solve for \( k \):
\[ k = \frac{2E}{A^2} \]
Now substituting the known values:
\[ k = \frac{2 \times 1 \, \text{J}}{(0.1 \, \text{m})^2} \]

### Step 4: Calculate \( A^2 \)
Calculating \( A^2 \):
\[ A^2 = (0.1 \, \text{m})^2 = 0.01 \, \text{m}^2 \]

### Step 5: Substitute \( A^2 \) back into the equation for \( k \)
Now substituting \( A^2 \) back into the equation for \( k \):
\[ k = \frac{2 \times 1}{0.01} = \frac{2}{0.01} = 200 \, \text{N/m} \]

### Step 6: Conclusion
Thus, the force constant \( k \) of the spring is:
\[ \boxed{200 \, \text{N/m}} \]