Calculate the work done on the tool by F⃗ of the tool is first moved out along the x-axis to the

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Calculate the work done on the tool by F⃗ of the tool is...

Calculate the work done on the tool by $\vec{F}$ of the tool is first moved out along the $x$-axis to the point $x=3.00 \mathrm{~m}, y=0$ and then moved parallel to the $y$-axis to $x=3.00 \mathrm{~m}, y=3.00 \mathrm{~m}$ (A) $67.5 \mathrm{~J}$ (B) $85 \mathrm{~J}$ (C) $102 \mathrm{~J}$ (D) $7.5 \mathrm{~J}$

Calculate the work done on the tool by $\vec{F}$ of the tool is first moved out along the $x$-axis to the point $x=3.00 \mathrm{~m}, y=0$ and then moved parallel to the $y$-axis to $x=3.00 \mathrm{~m}, y=3.00 \mathrm{~m}$
(A) $67.5 \mathrm{~J}$
(B) $85 \mathrm{~J}$
(C) $102 \mathrm{~J}$
(D) $7.5 \mathrm{~J}$

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Key Concepts

Work in Mechanics

Work in mechanics is the energy transfer that occurs when a force is applied over a displacement. It is calculated as the dot product of the force vector and the displacement vector, meaning only the component of the force in the direction of the displacement contributes to the work done.

Vector Dot Product

The vector dot product is used to measure how much one vector extends in the direction of another. In the context of work, calculating the dot product of the force and displacement vectors extracts the effective force component acting in the direction of movement, which is then multiplied by the displacement to determine the work done.

Line Integration of Force

Line integration allows for the calculation of work done when a force is applied along a path that may be segmented or non-linear. By integrating the dot product of the force vector and the differential displacement vector along the path, one can account for changes in both the magnitude and direction of the force throughout the movement.

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