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Work Energy Power

JEE Main level, numerical answer, medium

143 of 183

Question 143 of 183

A body of mass $2$ kg begins to move under the influence of time dependent force $\vec{F} = (2t\hat{i} + 6t^2\hat{j})$ N, where $\hat{i}$ and $\hat{j}$ are unit vectors along $x$ and $y$ -axis respectively. The power produced by the force at $t = 2$ s is _____ W.

Show answer and solution

Answer: 200

Given

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

N and

m = 2

kg.

Acceleration of the body is

\vec{a} = \dfrac{\vec{F}}{m} = \dfrac{2t\hat{i} + 6t^2\hat{j}}{2} = t\hat{i} + 3t^2\hat{j}

m/s

^2

.

Since the body begins to move from rest, its velocity at time

t

is given by

\vec{v} = \int \vec{a} dt = \int (t\hat{i} + 3t^2\hat{j}) dt = \dfrac{t^2}{2}\hat{i} + t^3\hat{j}

m/s.

At

t = 2

s, the force is

\vec{F} = 2(2)\hat{i} + 6(2)^2\hat{j} = 4\hat{i} + 24\hat{j}

N.

At

t = 2

s, the velocity is

\vec{v} = \dfrac{(2)^2}{2}\hat{i} + (2)^3\hat{j} = 2\hat{i} + 8\hat{j}

m/s.

Power produced by the force is

P = \vec{F} \cdot \vec{v} = (4\hat{i} + 24\hat{j}) \cdot (2\hat{i} + 8\hat{j})

P = (4 \times 2) + (24 \times 8) = 8 + 192 = 200

W.

Answer:

200

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