A particle of mass 100 gm moves in a potential well given by `U = 8 x^(2) - 4x + 400` joule. Find its acceleration at a distance of 25 cm from equilibrium in the positive direction

Options

A. (a)`4 ms^(-1)`
B. (b)` 40 ms^(-1)`
C. (c)` - 40 ms^(-1)`
D. (d)`-4 ms^(-1)`

Answer

To solve the problem step by step, we will follow these instructions:

### Step 1: Understand the potential energy function
The potential energy of the particle is given by:
\[ U(x) = 8x^2 - 4x + 400 \]
where \( x \) is the displacement from the equilibrium position.

### Step 2: Calculate the force from the potential energy
The force acting on the particle can be found by taking the negative derivative of the potential energy with respect to \( x \):
\[ F = -\frac{dU}{dx} \]

### Step 3: Differentiate the potential energy function
Now, we differentiate \( U(x) \):
\[ \frac{dU}{dx} = \frac{d}{dx}(8x^2 - 4x + 400) = 16x - 4 \]
Thus, the force becomes:
\[ F = - (16x - 4) = 4 - 16x \]

### Step 4: Find the equilibrium position
At equilibrium, the force is zero:
\[ 4 - 16x = 0 \]
Solving for \( x \):
\[ 16x = 4 \implies x = \frac{4}{16} = \frac{1}{4} \text{ m} = 25 \text{ cm} \]
This confirms that the equilibrium position is indeed at \( x = 25 \text{ cm} \).

### Step 5: Calculate the force at \( x = 50 \text{ cm} \)
We need to find the force when the particle is at \( x = 50 \text{ cm} \) (which is 25 cm from the equilibrium position in the positive direction):
\[ x = 50 \text{ cm} = 0.5 \text{ m} \]
Now substituting \( x = 0.5 \) into the force equation:
\[ F = 4 - 16(0.5) = 4 - 8 = -4 \text{ N} \]

### Step 6: Calculate the acceleration
Using Newton's second law, \( F = ma \), we can find the acceleration:
\[ a = \frac{F}{m} \]
Given the mass \( m = 100 \text{ gm} = 0.1 \text{ kg} \):
\[ a = \frac{-4 \text{ N}}{0.1 \text{ kg}} = -40 \text{ m/s}^2 \]

### Final Answer
The acceleration of the particle at a distance of 25 cm from the equilibrium position in the positive direction is:
\[ \boxed{-40 \text{ m/s}^2} \]