A body constrained to move along the z-axis of a coordinate system is subject to a constant force F given by F=- hat(i)+2 hat(j)+3 hat(k)N. Where hat(i), hat(j), hat(k) are unit vectors along the x,y and z axis of the system respectively. What is the work done by this force in moving the body a distance of 4m along the z axis?

Understand Work Done Definition: Understand the concept of work done by a force. Work done by a force is defined as the dot product of the force vector and the displacement vector. The formula for work done (W) is: W = F · d where F is the force vector, d is the displacement vector, and · represents the dot product.Identify Force and Displacement: Identify the components of the force vector and the displacement vector. The force vector F is given by F = - hat(i) + 2 hat(j) + 3 hat(k) N. The displacement vector d along the z-axis is d = 4 hat(k) m, since the body moves 4 meters along the z-axis and there is no movement along the x or y axes.Calculate Dot Product: Calculate the dot product of the force vector and the displacement vector. The dot product of two vectors is calculated by multiplying their corresponding components and then summing those products. Since the displacement is only along the z-axis, only the z-component of the force will do work. W = F · d = (- hat(i) + 2 hat(j) + 3 hat(k)) · (0 hat(i) + 0 hat(j) + 4 hat(k)) W = (0 * -1) + (0 * 2) + (3 * 4)Find Work Done: Perform the multiplication and addition to find the work done. W = 0 + 0 + 12 W = 12 Joules

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A body constrained to move along the
z-axis of a coordinate system is subject to a constant force
F given by
F=- hat(i)+2 hat(j)+3 hat(k)N. Where
hat(i), hat(j), hat(k) are unit vectors along the
x,y and
z axis of the system respectively. What is the work done by this force in moving the body a distance of
4m along the
z axis?

A body constrained to move along the $z$ -axis of a coordinate system is subject to a constant force $F$ given by $F=-\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}} N$ . Where $\hat{\mathbf{i}}, \hat{\mathbf{j}}, \hat{\mathbf{k}}$ are unit vectors along the $x, y$ and $z$ axis of the system respectively. What is the work done by this force in moving the body a distance of $4 \mathrm{~m}$ along the $\mathrm{z}$ axis?

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Q. A body constrained to move along the

z

-axis of a coordinate system is subject to a constant force

F

given by

F=-\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}} N

. Where

\hat{\mathbf{i}}, \hat{\mathbf{j}}, \hat{\mathbf{k}}

are unit vectors along the

x, y

and

z

axis of the system respectively. What is the work done by this force in moving the body a distance of

4 \mathrm{~m}

along the

\mathrm{z}

axis?

Understand Work Done Definition: Understand the concept of work done by a force. Work done by a force is defined as the dot product of the force vector and the displacement vector. The formula for work done $W$ is: $W = \mathbf{F} \cdot \mathbf{d}$ where $\mathbf{F}$ is the force vector, $\mathbf{d}$ is the displacement vector, and $\cdot$ represents the dot product.
Identify Force and Displacement: Identify the components of the force vector and the displacement vector. $\newline$ The force vector $\mathbf{F}$ is given by $\mathbf{F} = - \hat{i} + 2 \hat{j} + 3 \hat{k}$ N. $\newline$ The displacement vector $\mathbf{d}$ along the z-axis is $\mathbf{d} = 4 \hat{k}$ m, since the body moves $4$ meters along the z-axis and there is no movement along the x or y axes.
Calculate Dot Product: Calculate the dot product of the force vector and the displacement vector. $\newline$ The dot product of two vectors is calculated by multiplying their corresponding components and then summing those products. Since the displacement is only along the z-axis, only the z-component of the force will do work. $\newline$ $W = \mathbf{F} \cdot \mathbf{d} = (- \hat{i} + 2 \hat{j} + 3 \hat{k}) \cdot (0 \hat{i} + 0 \hat{j} + 4 \hat{k})$ $\newline$ $W = (0 \times -1) + (0 \times 2) + (3 \times 4)$
Find Work Done: Perform the multiplication and addition to find the work done. $\newline$ $W = 0 + 0 + 12$ $\newline$ $W = 12$ Joules