A block is constrained to move along x-axis under a force `F=-2x`. Here, F is in newton and x in metre. Find the work done by this force when the block is displaced from`x=2m to `x=-4` m.

Explanation

To find the work done by the force \( F = -2x \) when the block is displaced from \( x = 2 \, \text{m} \) to \( x = -4 \, \text{m} \), we can follow these steps:

### Step 1: Understand the Work Done Formula
The work done \( W \) by a force is given by the integral of the force over the displacement:
\[ W = \int F \, dx \]

### Step 2: Substitute the Force into the Integral
Given the force \( F = -2x \), we can substitute this into the work done formula:
\[ W = \int_{2}^{-4} (-2x) \, dx \]

### Step 3: Set Up the Integral
Now we need to evaluate the integral:
\[ W = -2 \int_{2}^{-4} x \, dx \]

### Step 4: Calculate the Integral
The integral of \( x \) is:
\[ \int x \, dx = \frac{x^2}{2} \]
Thus, we can evaluate the definite integral:
\[ W = -2 \left[ \frac{x^2}{2} \right]_{2}^{-4} \]

### Step 5: Evaluate the Limits
Now we need to evaluate this from \( x = 2 \) to \( x = -4 \):
\[ W = -2 \left( \frac{(-4)^2}{2} - \frac{(2)^2}{2} \right) \]
Calculating the squares:
\[ W = -2 \left( \frac{16}{2} - \frac{4}{2} \right) \]
This simplifies to:
\[ W = -2 \left( 8 - 2 \right) = -2 \times 6 = -12 \, \text{J} \]

### Final Answer
The work done by the force when the block is displaced from \( x = 2 \, \text{m} \) to \( x = -4 \, \text{m} \) is:
\[ W = -12 \, \text{J} \]

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