SATHEE: Work Energy and Power - Result Question 33

Work Energy and Power - Result Question 33

36. 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 $\hat{i}$ 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,

$m \frac{d \vec{v}}{d t} = 2 t \hat{i} + 3 t^{2} \hat{j} (m = 1 k g)$

$\Rightarrow \int_{0}^{\vec{v}} d \vec{v} = \int_{0}^{t} (2 t \hat{i} + 3 t^{2} \hat{j}) d t$

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

Power $P = \vec{F} \cdot \vec{v} (2 t \hat{i} + 3 t^{2} \hat{j}) \cdot (t^{2} \hat{i} + t^{3} \hat{j})$

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

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