Question

Calculate the work done in the following situations. A constant force \(\mathbf{F}=\langle 4,3,2\rangle\) (in newtons) moves an object from (0,0,0) to \((8,6,0) .\) (Distance is measured in meters.)

Short answer

Question: Calculate the work done by a constant force of 4 N in the x-direction, 3 N in the y-direction, and 2 N in the z-direction on an object as it moves from the origin (0,0,0) to the point (8,6,0). Answer: The work done by the force is 50 J (joules).

Step-by-step solution

Find the displacement vector

To find the displacement vector, subtract the initial position from the final position. Given the initial position is (0,0,0) and the final position is (8,6,0), the displacement vector is: \(\mathbf{r}_{f} - \mathbf{r}_{i} = \langle 8 - 0, 6 - 0, 0 - 0\rangle = \langle 8, 6, 0 \rangle\)

Calculate the dot product of force and displacement vectors

To find the work done by the force, we will calculate the dot product of the force vector and displacement vector. The dot product is defined as: \(\mathbf{F} \cdot \mathbf{r} = \langle F_{x}, F_{y}, F_{z} \rangle \cdot \langle r_{x}, r_{y}, r_{z} \rangle = F_{x} r_{x} + F_{y} r_{y} + F_{z} r_{z}\) So, in this case: \(\mathbf{F} \cdot \mathbf{r} = \langle 4, 3, 2 \rangle \cdot \langle 8, 6, 0 \rangle = 4 \times 8 + 3 \times 6 + 2 \times 0\)

Calculate the work done

Calculate the dot product: \(\mathbf{F} \cdot \mathbf{r} = 32 + 18 + 0 = 50\) The work done by the force is 50 J (joules).