Question

A body of mass \(2 \, \text{kg}\) begins to move under the action of a time-dependent force given by \(\vec{F} = (6t \, \hat{i} + 6t^2 \, \hat{j}) \, \text{N}\). The power developed by the force at the time \(t\) is given by:

• A. \((6t^4 + 9t^5) \, \text{W}\)
• B. \((3t^3 + 6t^5) \, \text{W}\)
• C. \((9t^5 + 6t^3) \, \text{W}\)
• D. \((9t^3 + 6t^5) \, \text{W}\)

Answer 1

The Correct Option is D

Given:

\[ \vec{F} = (6t \, \hat{i} + 6t^2 \, \hat{j}) \, \text{N} \]

The mass of the body is \( m = 2 \, \text{kg} \). According to Newton's second law:

\[ \vec{F} = m\vec{a} \implies \vec{a} = \frac{\vec{F}}{m} = \left(3t \, \hat{i} + 3t^2 \, \hat{j}\right) \, \text{m/s}^2 \]

The velocity \(\vec{v}\) is obtained by integrating the acceleration:

\[ \vec{v} = \int \vec{a} \, dt = \int \left(3t \, \hat{i} + 3t^2 \, \hat{j}\right) dt = \left(\frac{3t^2}{2} \, \hat{i} + t^3 \, \hat{j}\right) \, \text{m/s} \]

The power developed by the force is given by:

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

Calculating the dot product:

\[ P = (6t \, \hat{i} + 6t^2 \, \hat{j}) \cdot \left(\frac{3t^2}{2} \, \hat{i} + t^3 \, \hat{j}\right) \]

\[ P = 6t \cdot \frac{3t^2}{2} + 6t^2 \cdot t^3 \]

\[ P = 9t^3 + 6t^5 \, \text{W} \]