A body of mass 1 kg begins to move under the action of a time dependent force where and are unit vectors along x and y axis. What power will be developed by the force at the time ? [2016]

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Q.
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 along x and y axis. What power will be developed by the force at the time $t$ [2016]

1 $(2 t^{3} + 3 t^{4})$ $W$

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

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

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

Ans.
(2)

Here, $\vec{F} = (2 t \hat{i} + 3 t^{2} \hat{j}) N,$ $m = 1 kg$

Acceleration of the body, $\vec{a} = \frac{\vec{F}}{m} = \frac{(2 t \hat{i} + 3 t^{2} \hat{j}) N}{1 kg}$

Velocity of the body at time $t$ ,

$\vec{v} = \int \vec{a} d t = \int (2 t \hat{i} + 3 t^{2} \hat{j}) d t = t^{2} \hat{i} + t^{3} \hat{j}$ ${ms}^{- 1}$

$∴$ Power developed by the force at time $t$ ,

$P = \vec{F} \cdot \vec{v} = (2 t \hat{i} + 3 t^{2} \hat{j}) \cdot (t^{2} \hat{i} + t^{3} \hat{j}) W = (2 t^{3} + 3 t^{5}) W$

Solution

Q.
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 along x and y axis. What power will be developed by the force at the time $t$ [2016]

1 $(2 t^{3} + 3 t^{4})$ $W$

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

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

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

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Solution

Q. A body of mass 1 kg begins to move under the action of a time dependent force F→=(2t i^+3t2 j^)N, where i^ and j^ are unit vectors along x and y axis. What power will be developed by the force at the time t [2016]

1 (2t3+3t4)W

2 (2t3+3t5)W

3 (2t2+3t3)W

4 (2t2+4t4)W

Ans. (2) Here, F→=(2ti^+3t2j^)N, m=1kg Acceleration of the body, a→=F→m=(2ti^+3t2j^)N1kg Velocity of the body at time t, v→=∫a→dt=∫(2ti^+3t2j^) dt=t2i^+t3j^ ms-1 ∴ Power developed by the force at time t, P=F→·v→=(2ti^+3t2j^)·(t2i^+t3j^)W=(2t3+3t5)W

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Q.
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 along x and y axis. What power will be developed by the force at the time $t$ [2016]

1 $(2 t^{3} + 3 t^{4})$ $W$

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

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

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