Question

The area under the curve drawn between the force \(F\) and time \(t\) is

• A. torque
• B. impulse
• C. work done
• D. power
• E. kinetic energy

Hint:

The area under an F-x graph gives work done. The area under an F-t graph gives impulse.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
This question asks for the physical quantity represented by the area under a force-time (F-t) graph. This is a fundamental concept in mechanics related to Newton's second law.
Step 2: Key Formula or Approach:
From Newton's second law, force is the rate of change of momentum: \[ F = \frac{dp}{dt} \] where \(p\) is the momentum.
To find the total change in momentum over a time interval from \(t_1\) to \(t_2\), we can rearrange and integrate this equation: \[ dp = F \, dt \] \[ \int_{p_1}^{p_2} dp = \int_{t_1}^{t_2} F \, dt \] \[ p_2 - p_1 = \Delta p = \int_{t_1}^{t_2} F \, dt \] The quantity \(\Delta p\) (change in momentum) is defined as the impulse (J). The integral on the right-hand side represents the area under the F-t curve.
Step 3: Detailed Explanation:
The area under a curve of a function y(x) plotted against x is given by the definite integral \(\int y \, dx\).
In this case, the curve is force (F) plotted against time (t). So, the area under the curve is given by the integral \(\int F \, dt\).
As shown in Step 2, this integral is defined as the impulse imparted to the object. Therefore, the area under the force-time graph represents impulse.
Let's analyze the other options:
Work done is the area under a force-displacement (F-x) graph, \(W = \int F \, dx\).
Torque is a rotational force (\(\tau = r \times F\)).
Power is the rate of doing work (\(P = dW/dt = F \cdot v\)).
Kinetic energy is the energy of motion (\(K = \frac{1}{2}mv^2\)).
Step 4: Final Answer:
The area under the force-time graph is the impulse. This corresponds to option (B).