Question

A force \( F = (2 + x) \, \text{N} \) acts on a particle in the x-direction. The work done by this force during a displacement from \( x = 1.0 \, \text{m} \) to \( x = 2.0 \, \text{m} \) is?

• A. 2.1 J
• B. 2.5 J
• C. 3.5 J
• D. 4.5 J

Hint:

For variable forces, use the integral of force over displacement to calculate the work done.

Answer 1

The Correct Option is C

The work done by a variable force is given by the integral: \[ W = \int_{x_1}^{x_2} F(x) \, dx \] Substitute \( F(x) = (2 + x) \) and integrate from \( x_1 = 1.0 \, {m} \) to \( x_2 = 2.0 \, {m} \): \[ W = \int_{1.0}^{2.0} (2 + x) \, dx = \left[ 2x + \frac{x^2}{2} \right]_{1.0}^{2.0} \] Evaluating the integral: \[ W = \left( 2(2) + \frac{(2)^2}{2} \right) - \left( 2(1) + \frac{(1)^2}{2} \right) = \left( 4 + 2 \right) - \left( 2 + 0.5 \right) = 6 - 2.5 = 3.5 \, {J} \] Hence, the correct answer is (c).