At any instant of time, the total energy (E) of a simple pendulum is equal to the sum of its kinetic energy `(1/2 mv^(2))` and potential energy `(1/2kx^2)` where, m is the mass, v is the velocity, x is the displacement of the bob and k is a constant for the pendulum, the amplitude of oscillation of the pendulum is 10 cm and its total energy is 4 m J. Find k.

Options

A. `1.8 Nm^(-2)`
B. `0.8 Nm^(-2)`
C. `0.8 Nm^(-2)`
D. Data insufficient

Explanation

To solve the problem, we will follow these steps:

### Step 1: Understand the total energy of the pendulum
The total energy (E) of a simple pendulum is given by the sum of its kinetic energy (KE) and potential energy (PE):
\[ E = KE + PE \]
Where:
- Kinetic Energy \( KE = \frac{1}{2} mv^2 \)
- Potential Energy \( PE = \frac{1}{2} kx^2 \)

### Step 2: Identify the values given in the problem
From the problem, we know:
- The amplitude of oscillation \( A = 10 \, \text{cm} = 0.1 \, \text{m} \)
- The total energy \( E = 4 \, \text{mJ} = 4 \times 10^{-3} \, \text{J} \)

### Step 3: Determine the potential energy at maximum displacement
At maximum displacement (amplitude), the velocity \( v = 0 \) and all the energy is potential energy. Therefore, we can write:
\[ E = PE \]
\[ E = \frac{1}{2} k A^2 \]

### Step 4: Substitute the known values into the equation
Substituting the known values into the potential energy equation:
\[ 4 \times 10^{-3} = \frac{1}{2} k (0.1)^2 \]

### Step 5: Simplify the equation
Calculating \( (0.1)^2 \):
\[ (0.1)^2 = 0.01 \]
So the equation becomes:
\[ 4 \times 10^{-3} = \frac{1}{2} k (0.01) \]

### Step 6: Solve for k
Multiply both sides by 2 to eliminate the fraction:
\[ 8 \times 10^{-3} = k (0.01) \]
Now, divide both sides by \( 0.01 \):
\[ k = \frac{8 \times 10^{-3}}{0.01} \]
\[ k = 0.8 \, \text{N/m} \]

### Conclusion
The spring constant \( k \) is \( 0.8 \, \text{N/m} \).

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