A body of mass 2 kg begins to move under the action of a time dependent force given by The power developed by the force at the time is given by

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Solution

Q.
A body of mass 2 kg begins to move under the action of a time dependent force given by $\vec{F} = (6 t \hat{i} + 6 t^{2} \hat{j}) N .$ The power developed by the force at the time $t$ is given by [2024]

1 $(6 t^{4} + 9 t^{5}) W$

2 $(3 t^{5} + 6 t^{5}) W$

3 $(9 t^{5} + 6 t^{3}) W$

4 $(9 t^{3} + 6 t^{5}) W$

Ans.
(4)

   $\vec{F} = (6 t \hat{i} + 6 t^{2} \hat{j}) N$

   $\vec{F} = m \vec{a} = (6 t \hat{i} + 6 t^{2} \hat{j})$

   $\vec{a} = \frac{\vec{F}}{m} = (3 t \hat{i} + 3 t^{2} \hat{j})$

   $\vec{v} = \int_{0}^{t} \vec{a} d t = \frac{3 t^{2}}{2} \hat{i} + t^{3} \hat{j}$

$P = \vec{F} \cdot \vec{v} = (9 t^{3} + 6 t^{5}) W$

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