Question

A body of mass 1 kg starts moving from rest under the action of a force which varies with displacement as $F=2x+5$ (in newtons). The work done by this force to displace the body from $x=0$ to $x=2$ m is:

• A. 8 J
• B. 10 J
• C. 12 J
• D. 14 J

Hint:

For linear force $F=mx+c$, Work is also (Average Force $\times$ Displacement). $F_{avg} = (5 + 9)/2 = 7$; $W = 7 \times 2 = 14$ J.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
When force is a function of displacement, work done is calculated by integrating the force with respect to displacement over the given interval.
Step 2: Key Formula or Approach:
$W = \int_{x_1}^{x_2} F \, dx$.
Step 3: Detailed Explanation:
\[ W = \int_{0}^{2} (2x + 5) \, dx \] Integrating the expression: \[ W = \left[ \frac{2x^2}{2} + 5x \right]_0^2 \] \[ W = [x^2 + 5x]_0^2 \] Substituting the limits: \[ W = (2^2 + 5(2)) - (0^2 + 5(0)) \] \[ W = 4 + 10 = 14 \text{ J} \]
Step 4: Final Answer:
The work done by the force is 14 J.