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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
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B
33J
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C
83J
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D
45J
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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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