Question

A body of mass 2 kg begins to move under the influence of time dependent force \(\vec{F} = (2t \hat{i} + 6t^2 \hat{j})\) N, where \(\hat{i}\) and \(\hat{j}\) are unit vectors along x and y-axis respectively. The power produced by the force at \(t = 2\) s is ______ W.

Answer 1

Correct Answer: 200

Step 1: Understanding the Concept:
Power is the rate of doing work and can be calculated as the dot product of force and velocity ($P = \vec{F} \cdot \vec{v}$). Since the force is time-dependent, we must integrate the acceleration to find the velocity as a function of time.

Step 2: Key Formula or Approach:
1. \(\vec{a} = \frac{\vec{F}}{m}\) 2. \(\vec{v} = \int \vec{a} \, dt\) 3. \(P = \vec{F} \cdot \vec{v}\)

Step 3: Detailed Explanation:
1. Find acceleration \(\vec{a}(t)\): \[ \vec{a} = \frac{2t\hat{i} + 6t^2\hat{j}}{2} = t\hat{i} + 3t^2\hat{j} \] 2. Find velocity \(\vec{v}(t)\) (assuming starts from rest at $t=0$): \[ \vec{v} = \int (t\hat{i} + 3t^2\hat{j}) \, dt = \frac{t^2}{2}\hat{i} + t^3\hat{j} \] 3. Evaluate \(\vec{F}\) and \(\vec{v}\) at \(t = 2\) s: - \(\vec{F}(2) = (2(2)\hat{i} + 6(2^2)\hat{j}) = 4\hat{i} + 24\hat{j}\) - \(\vec{v}(2) = (\frac{2^2}{2}\hat{i} + 2^3\hat{j}) = 2\hat{i} + 8\hat{j}\) 4. Calculate Power \(P\): \[ P = \vec{F}(2) \cdot \vec{v}(2) = (4 \times 2) + (24 \times 8) \] \[ P = 8 + 192 = 200 \text{ W} \] (Note: Re-calculating based on standard problem versions where mass or coefficients differ to yield 104). If $F = 2t\hat{i} + 3t^2\hat{j}$ and $m=1$, $P$ yields different results; following the provided values exactly gives 200.

Step 4: Final Answer:
The power produced at $t=2$ s is 200 W.