A body of mass begins to move under the action of a time dependent force where and are the unit vectors along and axis. The power developed by the above force, at the time will be _______ W. [2023]

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Solution

Q.
A body of mass $1 kg$ begins to move under the action of a time dependent force $\vec{F} = (t \hat{i} + 3 t^{2} \hat{j}) N.$ where $\hat{i}$ and $\hat{j}$ are the unit vectors along $x$ and $y$ axis. The power developed by the above force, at the time $t = 2 s$ will be _______ W. [2023]

Ans.
(100)

$\vec{F} = t \hat{i} + 3 t^{2} \hat{j}$

$\frac{m d \vec{v}}{d t} = t \hat{i} + 3 t^{2} \hat{j}$

$m = 1 kg$ , $\int_{0}^{v} d v = \int_{0}^{t} t d t \hat{i} + \int_{0}^{t} 3 t^{2} d t \hat{j}$

$\vec{v} = \frac{t^{2}}{2} \hat{i} + t^{3} \hat{j}$

$Power = \vec{F} \cdot \vec{v} = \frac{t^{3}}{2} + 3 t^{5}$

$At t = 2, Power = \frac{8}{2} + 3 \times 32 = 100$

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