Questions: A constant force F=-1 i+2 j-1 k is applied to an object that is moving along a straight line from the point (-5,4,4) to the point (3,-4,-4). Find the work done if the distance is measured in meters and the force in newtons. Include units in your answer. (Note, units are case sensitive. Clicking on the link units will give a list of units.) Answer =

A constant force F=-1 i+2 j-1 k is applied to an object that is moving along a straight line from the point (-5,4,4) to the point (3,-4,-4). Find the work done if the distance is measured in meters and the force in newtons. Include units in your answer. (Note, units are case sensitive. Clicking on the link units will give a list of units.)

Answer =

Transcript text: A constant force $\mathbf{F}=-1 \mathbf{i}+2 \mathbf{j}-1 \mathbf{k}$ is applied to an object that is moving along a straight line from the point $(-5,4,4)$ to the point $(3,-4,-4)$. Find the work done if the distance is measured in meters and the force in newtons. Include units in your answer. (Note, units are case sensitive. Clicking on the link units will give a list of units.) Answer $=$

Solution

Solution Steps

Step 1: Determine the Displacement Vector

The displacement vector $\mathbf{d}$ is found by subtracting the initial position vector from the final position vector. The initial position is $(-5, 4, 4)$ and the final position is $(3, -4, -4)$.

\[ \mathbf{d} = (3 - (-5))\mathbf{i} + (-4 - 4)\mathbf{j} + (-4 - 4)\mathbf{k} = 8\mathbf{i} - 8\mathbf{j} - 8\mathbf{k} \]

Step 2: Calculate the Work Done

The work done $W$ by a constant force is given by the dot product of the force vector $\mathbf{F}$ and the displacement vector $\mathbf{d}$.

\[ W = \mathbf{F} \cdot \mathbf{d} = (-1\mathbf{i} + 2\mathbf{j} - 1\mathbf{k}) \cdot (8\mathbf{i} - 8\mathbf{j} - 8\mathbf{k}) \]

Calculate the dot product:

\[ W = (-1)(8) + (2)(-8) + (-1)(-8) = -8 - 16 + 8 = -16 \]