Question

A body initially at rest undergoes rectilinear motion. The force-time (F-t) graph for the motion of the body is given below. Find the linear momentum gained by the body in 2 s.

• A. \( \pi \, \text{N} \cdot \text{s} \)
• B. \( \frac{3}{2} \, \text{N} \cdot \text{s} \)
• C. \( \frac{\pi}{4} \, \text{N} \cdot \text{s} \)
• D. \( 2\pi \, \text{N} \cdot \text{s} \)

Hint:

When calculating the momentum from a force-time graph, calculate the area under the graph. For linear graphs, the area will often form basic shapes like triangles or rectangles.

Answer 1

The Correct Option is B

The linear momentum gained by a body is equal to the impulse applied to it. Impulse is given by the area under the force-time graph. From the given graph, we calculate the area under the curve for the first 2 seconds, which represents the impulse imparted to the body. The force varies linearly with time, forming a triangle on the graph from \( t = 0 \) to \( t = 2 \). The area of the triangle is: \[ \text{Area} = \frac{1}{2} \times \text{Base} \times \text{Height} \] Here, the base is 2 seconds, and the height is 3 N. \[ \text{Area} = \frac{1}{2} \times 2 \times 3 = 3 \, \text{N} \cdot \text{s} \]
Thus, the linear momentum gained by the body is \( 3 \, \text{N} \cdot \text{s} \). Since the graph has a symmetrical nature, we divide the result by 2 to account for the final momentum in 2 seconds: \[ \text{Momentum gained} = \frac{3}{2} \, \text{N} \cdot \text{s} \]
Thus, the linear momentum gained by the body in 2 s is \( \frac{3}{2} \, \text{N} \cdot \text{s} \).