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

To determine the power exerted by a time-varying force on an object, follow these steps:

1. The applied force is defined by the vector function:

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

1. .
2. The object's velocity, initially zero, is obtained by integrating its acceleration. The acceleration is given by:

\[\vec{a} = \frac{\vec{F}}{m} = \left(\frac{6t}{2} \, \hat{i} + \frac{6t^2}{2} \, \hat{j}\right) \, \text{m/s}^2 = (3t \, \hat{i} + 3t^2 \, \hat{j}) \, \text{m/s}^2\]

1. .
2. Integrate the acceleration with respect to time to find the velocity:

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

1. .

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

• .

1. Calculate the power developed by the force using the formula:

\[\text{Power} = \vec{F} \cdot \vec{v}\]

1. (the dot product of the force and velocity vectors).
   • Substitute the expressions for \( \vec{F} \) and \( \vec{v} \):

\[\vec{F} \cdot \vec{v} = (6t) \left(\frac{3t^2}{2}\right) + (6t^2) (t^3)\]

• Simplify the expression:

\[\vec{F} \cdot \vec{v} = 9t^3 + 6t^5\]

1. The power developed by the force at any time \( t \) is:

\[(9t^3 + 6t^5) \, \text{W}\]

1. .

Therefore, the power developed is: (9t³ + 6t⁵) W.