First slide

Power

Question

A body of mass $1 k g$ begins to move under the action of a time dependent force $\vec{F} = (2 t \hat{i} + 3 t^{2} \hat{j}) N,$ where $\hat{i} a n d \hat{j}$ are unit vectors along $x$
and $y$ axis. What power will be developed by the force at the time $t$ ?

Moderate

A

$(2 t^{3} + 3 t^{4}) W$
B

$(2 t^{3} + 3 t^{5}) W$
C

$(2 t^{2} + 3 t^{3}) W$
D

$(2 t^{2} + 4 t^{4}) W$

Solution

$Here, \vec{F} = (2 t \hat{i} + 3 t^{2} \hat{j}) N, m = 1 kg$
$Velocity of the body at time t$
$\vec{v} = \int \vec{a} d t = \int (2 t \hat{i} + 3 t^{2} \hat{j}) d t = t^{2} \hat{i} + t^{3} \hat{j} m s^{- 1}$
$∴ Power developed by the force at time t,$
$P = \vec{F} \cdot \vec{v} = (2 t \hat{i} + 3 t^{2} \hat{j}) \cdot (t^{2} \hat{i} + t^{3} \hat{j}) W = (2 t^{3} + 3 t^{5}) W$

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