Work Energy and Power - Result Question 33
36. A body of mass $1 kg$ begins to move under the action of a time dependent force $\vec{F}=(2 t \hat{i}+3 t^{2} \hat{j}) N$, where $\hat{i}$ and $\hat{j}$ are unit vectors alogn $x$ and $y$-axis. What power will be developed by the force at the time $t$ ? [2016]
(a) $(2 t^{2}+3 t^{3}) W$
(b) $(2 t^{2}+4 t^{4}) W$
(c) $(2 t^{3}+3 t^{4}) W$
(d) $(2 t^{3}+3 t^{5}) W$
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Answer:
Correct Answer: 36. (d)
Solution:
- (d) Given force $\vec{F}=2 t \hat{i}+3 t^{2} \hat{j}$
According to Newton’s second law of motion,
$m \frac{d \vec{v}}{d t}=2 t \hat{i}+3 t^{2} \hat{j} \quad(m=1 kg)$
$\Rightarrow \int_0^{\vec{v}} d \vec{v}=\int_0^{t}(2 t \hat{i}+3 t^{2} \hat{j}) d t$
$\Rightarrow \vec{v}=t^{2} \hat{i}+t^{3} \hat{j}$
Power $P=\vec{F} \cdot \vec{v}(2 t \hat{i}+3 t^{2} \hat{j}) \cdot(t^{2} \hat{i}+t^{3} \hat{j})$
$=(2 t^{3}+3 t^{5}) W$
