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Chapter 11: Problem 40

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

Expert verified
Question: Calculate the work done in moving an object from the initial point (0,0,0) to the final point (8,6,0), given the force vector F = 鉄�4,3,2鉄�. Answer: The work done is 50 Joules.

Step by step solution

01

Calculate the displacement vector

To find the displacement vector, we need to subtract the initial position from the final position. Let's call the displacement vector \(\mathbf{d}\). So we have: \(\mathbf{d} = (8,6,0) - (0,0,0) = \langle 8,6,0 \rangle\)
02

Calculate the dot product of the force and displacement vectors

Now, we need to find the dot product of the force vector \(\mathbf{F}\) and the displacement vector \(\mathbf{d}\). The formula for the dot product is: \(\mathbf{F} \cdot \mathbf{d} = F_xd_x + F_yd_y + F_zd_z\) Using the given force vector \(\mathbf{F} = \langle 4,3,2 \rangle\) and the calculated displacement vector \(\mathbf{d} = \langle 8,6,0 \rangle\), the dot product can be calculated as follows: \(\mathbf{F} \cdot \mathbf{d} = (4)(8) + (3)(6) + (2)(0) = 32 + 18 + 0 = 50\)
03

Determine the work done

The dot product of the force and displacement vectors is equal to the work done, so we have: \(W = \mathbf{F} \cdot \mathbf{d} = 50\) Therefore, the work done in moving the object from \((0,0,0)\) to \((8,6,0)\) is \(50\ \text{Joules}\).

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