Question

# An object has a displacement from position vector $$\vec{r_1} = (2\hat{i}+ 3\hat{j} )m$$to  $$\vec{r_2} = (4\hat{i}+ 6\hat{j})m$$ under a force $$\vec{F} = (3x^2\hat{i} + 2y \hat{j} )N,$$ then work done by the force is:

A
24J
B
33J
C
83J
D
45J

Solution

## The correct option is C $$83J$$$$\begin{array}{l} w=\int _{ \overrightarrow { { r_{ 1 } } } }^{ \overrightarrow { { r_{ 2 } } } }{ \overrightarrow { F } \left( { dx\widehat { i } +dy\widehat { j } } \right) } \\ =\int _{ \overrightarrow { { r_{ 1 } } } }^{ \overrightarrow { { r_{ 2 } } } }{ \left( { 3{ x^{ 2 } }\widehat { i } +2y\widehat { j } } \right) }\left( { dx\widehat { i } +dy\widehat { j } } \right) \\ =\int _{ 2 }^{ 4 }{ 3{ x^{ 2 } }dx+ }\int _{ 3 }^{ 6 }{ 3y\, dy } \\ =\left[ { { x^{ 3 } } } \right] _{ 2 }^{ 4 }+\left[ { { y^{ 2 } } } \right] _{ 3 }^{ 6 } \\ =\left( { 64-8 } \right) +\left( { 36-9 } \right) \\ =56+27 \\ =83\, J \end{array}$$$$\therefore$$ Option $$C$$ is correct.Physics

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