Question

A body of mass \(1 \, \text{kg}\) begins to move under the action of a time-dependent force \[ \vec{F} = (t \, \hat{i} + 3t^2 \, \hat{j}) \, \text{N}, \] where \(\hat{i}\) and \(\hat{j}\) are unit vectors along \(x\) and \(y\) axes. The power developed by the above force, at the time \(t = 2 \, \text{s}\), will be _____ W.

Hint:

The power developed by a force is \[ P = \vec{F} \cdot \vec{v}. \] Ensure both \(\vec{F}\) and \(\vec{v}\) are evaluated at the same instant of time.

Answer 1

Correct Answer: 100

The force acting on the body is:

\[ \vec{F} = t \, \hat{i} + 3t^2 \, \hat{j} \]

The acceleration is given by Newton’s second law:

\[ \vec{a} = \frac{\vec{F}}{m} = \vec{F} \quad (\text{since } m = 1 \, \text{kg}) \]

The velocity is obtained by integrating acceleration:

\[ \vec{v} = \int \vec{a} \, dt = \int (t \, \hat{i} + 3t^2 \, \hat{j}) \, dt = \frac{t^2}{2} \, \hat{i} + t^3 \, \hat{j} \]

At \(t = 2 \, \text{s}\):

\[ \vec{v} = \frac{2^2}{2} \, \hat{i} + 2^3 \, \hat{j} = 2 \, \hat{i} + 8 \, \hat{j} \]

The power is given by:

\[ P = \vec{F} \cdot \vec{v} \]

Substitute \(\vec{F} = 2 \, \hat{i} + 3 \cdot 2^2 \, \hat{j} = 2 \, \hat{i} + 12 \, \hat{j}\) and \(\vec{v} = 2 \, \hat{i} + 8 \, \hat{j}\):

\[ P = (2 \, \hat{i} + 12 \, \hat{j}) \cdot (2 \, \hat{i} + 8 \, \hat{j}) = (2 \cdot 2) + (12 \cdot 8) = 4 + 96 = 100 \, \text{W} \]

Thus, the power at \(t = 2 \, \text{s}\) is \(100 \, \text{W}\).

Answer 2

The correct answer is 100. F=ti^+3t2j^​ dtmdv​=ti^+3t2j^​ m=1kg,0∫v​dv=0∫t​tdti^+0∫t​3t2dtj^​ v=2t2​i^+t3j^​ Power =F⋅V=2t3​+3t5 At t=2, power =28​+3×32 =100