The potential energy of particle of mass 1kg moving along the x-axis is given by `U(x) = 16(x^(2) - 2x)` J, where x is in meter. Its speed at x=1 m is` 2 m//s`. Then,

Options

A. the motion of the particle is uniformly accelerated
B. the motion of the particle is periodic but not simple harmonic.
C. the motion of the particle is simple harmonic having equilibrium position at x=1m
D. kinetic energy of the particle is conserved

Answer

To solve the problem step by step, we will analyze the potential energy function and derive the necessary information about the motion of the particle.

### Step 1: Understand the Potential Energy Function
The potential energy \( U(x) \) of the particle is given by:
\[ U(x) = 16(x^2 - 2x) \, \text{J} \]
This function describes how the potential energy changes with the position \( x \).

### Step 2: Calculate the Force Acting on the Particle
The force \( F \) acting on the particle can be found using the negative gradient of the potential energy:
\[ F = -\frac{dU}{dx} \]
First, we differentiate \( U(x) \):
\[ \frac{dU}{dx} = 16 \frac{d}{dx}(x^2 - 2x) = 16(2x - 2) = 32(x - 1) \]
Thus, the force is:
\[ F = -32(x - 1) \]

### Step 3: Identify the Equilibrium Position
The equilibrium position occurs when the force is zero:
\[ F = 0 \implies -32(x - 1) = 0 \implies x = 1 \, \text{m} \]
This indicates that at \( x = 1 \, \text{m} \), the particle experiences no net force.

### Step 4: Determine the Nature of the Motion
Since the force is proportional to the displacement from the equilibrium position (which is \( x = 1 \)), we can conclude that the motion is simple harmonic motion (SHM). The force can be expressed as:
\[ F = -k(x - x_0) \]
where \( k = 32 \) and \( x_0 = 1 \).

### Step 5: Analyze the Motion Characteristics
In SHM, the motion is periodic, and the particle oscillates around the equilibrium position. The speed of the particle at \( x = 1 \, \text{m} \) is given as \( 2 \, \text{m/s} \).

### Step 6: Evaluate the Options
1. **The motion of the particle is uniformly accelerated.** - **False**: The acceleration changes with position.
2. **The motion of the particle is periodic but not simple harmonic.** - **False**: It is indeed simple harmonic motion.
3. **The kinetic energy of the particle is conserved.** - **False**: In SHM, the total mechanical energy (kinetic + potential) is conserved, not just kinetic energy.

### Conclusion
The correct answer is that the motion of the particle is simple harmonic motion with an equilibrium position at \( x = 1 \, \text{m} \).

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