Question

A force \( \mathbf{F} = (2 + 3x) \hat{i})\) acts on a particle in the \( x \)-direction where \( F \) is in newton and \( x \) is in meter. The work done by this force during a displacement from \( x = 0 \) to \( x = 4 \, \text{m} \) is _____ J.

Hint:

To calculate the work done by a variable force, integrate the force with respect to displacement over the given limits.

Answer 1

Correct Answer: 32

The work done by a force is given by the integral: \[ w = \int_{x_1}^{x_2} F(x) \, dx \] Where \( F(x) = 2 + 3x \). Substituting into the integral: \[ w = \int_0^4 (2 + 3x) \, dx \] Solving this: \[ w = \int_0^4 2 \, dx + \int_0^4 3x \, dx = [2x]_0^4 + \left[ \frac{3x^2}{2} \right]_0^4 \] \[ w = (2 \times 4) + \left( \frac{3 \times 4^2}{2} \right) = 8 + \left( \frac{3 \times 16}{2} \right) = 8 + 24 = 32 \, \text{J} \] Thus, the work done is 32 J.