Potential energy of a particle moving along x-axis under the action of only conservative force is given as `U=10+4cos(4pi x)`. Here, U is in Joule and x in metres. Total mechanial energy of the particle is 16 J. Choose the correct option.

Options

A. At x = 1.25 m, particle is at equilibrium position
B. Maximum kinetic energy of the particle is 20 J
C. Both (a) and (b) are correct
D. Both (a) and (b) and wrong

Answer

To solve the problem, we need to analyze the potential energy function and the total mechanical energy of the particle.

### Given:
- Potential energy \( U = 10 + 4 \cos(4 \pi x) \) (in Joules)
- Total mechanical energy \( E = 16 \) J

### Step 1: Write the expression for total mechanical energy
The total mechanical energy \( E \) is the sum of kinetic energy \( K \) and potential energy \( U \):
\[ E = K + U \]
From this, we can express the kinetic energy as:
\[ K = E - U \]
Substituting the given values:
\[ K = 16 - (10 + 4 \cos(4 \pi x)) \]
This simplifies to:
\[ K = 6 - 4 \cos(4 \pi x) \]

### Step 2: Find the maximum kinetic energy
The kinetic energy \( K \) will be maximum when \( \cos(4 \pi x) \) is at its minimum value, which is -1. Therefore, we substitute \( \cos(4 \pi x) = -1 \):
\[ K_{\text{max}} = 6 - 4(-1) = 6 + 4 = 10 \, \text{J} \]

### Step 3: Determine the equilibrium position
The equilibrium position occurs when the force acting on the particle is zero. The force \( F \) can be derived from the potential energy as:
\[ F = -\frac{dU}{dx} \]
Calculating the derivative:
\[ U = 10 + 4 \cos(4 \pi x) \]
\[ \frac{dU}{dx} = -4 \cdot 4 \pi \sin(4 \pi x) = -16 \pi \sin(4 \pi x) \]
Thus, the force is:
\[ F = 16 \pi \sin(4 \pi x) \]
Setting the force to zero for equilibrium:
\[ 16 \pi \sin(4 \pi x) = 0 \]
This implies:
\[ \sin(4 \pi x) = 0 \]
The sine function is zero when:
\[ 4 \pi x = n \pi \quad (n = 0, 1, 2, \ldots) \]
This simplifies to:
\[ x = \frac{n}{4} \]

### Step 4: Check the given options
We need to check if \( x = 0.125 \) (which is \( \frac{1}{8} \)) corresponds to an equilibrium position:
\[ x = 0.125 \implies 4 \pi x = 4 \pi \cdot 0.125 = \pi \]
Since \( \sin(\pi) = 0 \), this means \( x = 0.125 \) is indeed an equilibrium position.

### Conclusion
The correct option is that the particle is at equilibrium at \( x = 0.125 \).