Question

The force, $F = (2x^2 + 5x + 4)$ N is acting on a body. If the body moves from $x = -1$ m to $x = +1$ m in the direction of force, then the work done by the force is

• A. 3.43 J
• B. 9.33 J
• C. 6.86 J
• D. 10.3 J

Hint:

When force is a function of position, always integrate force over the displacement to find work.

Answer 1

The Correct Option is B

We are given force as a function of position: $F(x) = 2x^2 + 5x + 4$
To find work done, we integrate force over the displacement: $W = \int_{-1}^{1} F(x) \, dx = \int_{-1}^{1} (2x^2 + 5x + 4) \, dx$
Break the integral: $\int_{-1}^{1} 2x^2 \, dx + \int_{-1}^{1} 5x \, dx + \int_{-1}^{1} 4 \, dx$
$= 2 \int_{-1}^{1} x^2 dx + 5 \int_{-1}^{1} x dx + 4 \int_{-1}^{1} dx$
$= 2\left[\dfrac{x^3}{3}\right]_{-1}^{1} + 5\left[\dfrac{x^2}{2}\right]_{-1}^{1} + 4[x]_{-1}^{1}$
$= 2\left(\dfrac{1^3 - (-1)^3}{3}\right) + 5(0) + 4(1 - (-1))$
$= 2\left(\dfrac{1 + 1}{3}\right) + 0 + 4(2) = 2\left(\dfrac{2}{3}\right) + 8 = \dfrac{4}{3} + 8 = \dfrac{28}{3} \approx 9.33$ J