Question 227: Work, Energy and Power & Circular Motion

Question

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

Accepted Answer

Given,
Mass of body: $m = 2\,	ext{kg}$
Force:
$$ \vec{F} = \left(2t\,\hat{i} + 6t^2\,\hat{j}ight)\,	ext{N} $$
We have to find the power at $t=2\,	ext{s}$.
Power produced by a force is
$$ P = \vec{F} \cdot \vec{v} $$
So first we need velocity at time $t=2\,	ext{s}$.
Step 1: Find acceleration
Using NewtonÃ¢ÂÂs second law,
$$ \vec{F} = m\vec{a} $$
So,
$$ \vec{a} = \frac{\vec{F}}{m} = \frac{1}{2}\left(2t\,\hat{i} + 6t^2\,\hat{j}ight) $$
$$ \vec{a} = t\,\hat{i} + 3t^2\,\hat{j} $$
Step 2: Find velocity
Since the body "begins to move", initial velocity is zero.
Now integrate acceleration:
For $x$-component,
$$ a_x = t $$
$$ v_x = \int t\,dt = \frac{t^2}{2} $$
For $y$-component,
$$ a_y = 3t^2 $$
$$ v_y = \int 3t^2\,dt = t^3 $$
Hence,
$$ \vec{v} = \frac{t^2}{2}\,\hat{i} + t^3\,\hat{j} $$
At $t=2\,	ext{s}$,
$$ \vec{v}(2) = \frac{2^2}{2}\,\hat{i} + 2^3\,\hat{j} = 2\,\hat{i} + 8\,\hat{j} $$
Step 3: Find force at $t=2\,	ext{s}$
$$ \vec{F}(2) = \left(2(2)\,\hat{i} + 6(2)^2\,\hat{j}ight) $$
$$ \vec{F}(2) = 4\,\hat{i} + 24\,\hat{j} $$
Step 4: Find power
$$ P = \vec{F} \cdot \vec{v} $$
$$ P = (4\,\hat{i} + 24\,\hat{j}) \cdot (2\,\hat{i} + 8\,\hat{j}) $$
$$ P = 4 \cdot 2 + 24 \cdot 8 $$
$$ P = 8 + 192 = 200 $$
Therefore, the power produced at $t=2\,	ext{s}$ is
$$ \boxed{200\ 	ext{W}} $$

Work, Energy and Power & Circular Motion

Integer / Numeric Answer Type