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, we start with the given potential energy function of the particle:

**Step 1: Write down the potential energy function.**
\[ U(x) = 16(x^2 - 2x) \, \text{J} \]

**Step 2: Differentiate the potential energy function to find the force.**
The force \( F \) acting on the particle can be found using the relation:
\[ F = -\frac{dU}{dx} \]

Differentiating \( U(x) \):
\[ \frac{dU}{dx} = \frac{d}{dx}(16x^2 - 32x) = 32x - 32 \]
Thus, the force is:
\[ F(x) = - (32x - 32) = -32x + 32 \]

**Step 3: Relate force to mass and acceleration.**
Using Newton's second law, we have:
\[ F = ma \]
where \( m = 1 \, \text{kg} \). Therefore:
\[ ma = -32x + 32 \]
\[ a = -32x + 32 \]

**Step 4: Analyze the acceleration.**
The acceleration \( a \) can be expressed as:
\[ a = -32(x - 1) \]
This indicates that the acceleration is proportional to the displacement from the equilibrium position \( x = 1 \).

**Step 5: Determine the nature of motion.**
Since the acceleration is proportional to the negative displacement, this indicates that the motion is simple harmonic. The equilibrium position is at \( x = 1 \, \text{m} \).

**Step 6: Evaluate the options given in the question.**
- The motion is periodic but not simple harmonic: **Incorrect** (it is simple harmonic).
- The motion is uniformly accelerated: **Incorrect** (acceleration depends on \( x \)).
- The motion is simple harmonic with an equilibrium position at \( x = 1 \, \text{m} \): **Correct**.
- Kinetic energy of the particle is conserved: **Incorrect** (total energy is conserved, not kinetic energy alone).

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