Question

A force \( (3x^2 + 2x - 5) \, \text{N} \) displaces a body from \( x = 2 \, \text{m} \) to \( x = 4 \, \text{m} \). The work done by this force is _________ J.

Answer 1

Correct Answer: 58

To determine the work performed by the force \( F(x) = 3x^2 + 2x - 5 \) during a displacement from \( x = 2 \) m to \( x = 4 \) m, we employ the work integral:

\( W = \int_{2}^{4} F(x) \, dx = \int_{2}^{4} (3x^2 + 2x - 5) \, dx \)

First, we compute the antiderivative:

\(\int (3x^2 + 2x - 5) \, dx = \int 3x^2 \, dx + \int 2x \, dx - \int 5 \, dx = x^3 + x^2 - 5x + C\)

Next, we evaluate the definite integral using the limits from 2 to 4:

\(W = [x^3 + x^2 - 5x]_{2}^{4} = [(4^3 + 4^2 - 5 \times 4) - (2^3 + 2^2 - 5 \times 2)]\)

The individual terms are calculated as follows:

\(4^3 = 64, \, 4^2 = 16\)

\(2^3 = 8, \, 2^2 = 4\)

Substituting these values into the expression:

\(= (64 + 16 - 20) - (8 + 4 - 10)\)

\(= 60 - 2\)

\(= 58 \, \text{J}\)

The total work done by the force is \(58\) J. This value is consistent with the specified range of (58,58).