The potential energy of a particle is given by formula `U=100-5x + 100x^(2), where `U` and 'x' are in SI unit .if mass of particle is 0.1 Kg then find the magnitude of its acceleration

Options

A. At `0.05m` from the origin is 50 ms^(2).
B. At 0.05 m from the mean position is 100 ms^(2).
C. At 0.05 m from the origin is 150 ms^(2).
D. At `0.05 m` from the mean position is `200 ms^(-1)`

Explanation

To solve the problem, we need to find the magnitude of the acceleration of a particle given its potential energy function and its mass. The potential energy \( U \) is given by:

\[ U = 100 - 5x + 100x^2 \]

where \( U \) is in Joules and \( x \) is in meters. The mass of the particle is \( m = 0.1 \, \text{kg} \).

### Step 1: Calculate the Force from Potential Energy
The force \( F \) acting on the particle can be derived from the potential energy using the relation:

\[ F = -\frac{dU}{dx} \]

We need to differentiate \( U \) with respect to \( x \):

\[ \frac{dU}{dx} = \frac{d}{dx}(100 - 5x + 100x^2) = 0 - 5 + 200x = -5 + 200x \]

Thus, the force is:

\[ F = -\left(-5 + 200x\right) = 5 - 200x \]

### Step 2: Calculate Acceleration
Using Newton's second law, the acceleration \( a \) can be calculated as:

\[ a = \frac{F}{m} \]

Substituting the expression for force:

\[ a = \frac{5 - 200x}{0.1} \]

This simplifies to:

\[ a = 50 - 2000x \]

### Step 3: Evaluate Acceleration at Specific Positions
We need to find the acceleration at \( x = 0.05 \, \text{m} \):

\[ a = 50 - 2000(0.05) \]

Calculating this gives:

\[ a = 50 - 100 = -50 \, \text{m/s}^2 \]

The magnitude of acceleration is:

\[ |a| = 50 \, \text{m/s}^2 \]

### Step 4: Check Other Positions
Next, we check the acceleration at \( x = -0.05 \, \text{m} \):

\[ a = 50 - 2000(-0.05) = 50 + 100 = 150 \, \text{m/s}^2 \]

### Summary of Results
1. At \( x = 0.05 \, \text{m} \), the magnitude of acceleration is \( 50 \, \text{m/s}^2 \).
2. At \( x = -0.05 \, \text{m} \), the magnitude of acceleration is \( 150 \, \text{m/s}^2 \).

### Final Answer
The magnitude of the particle's acceleration at \( x = 0.05 \, \text{m} \) is \( 50 \, \text{m/s}^2 \).

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