Problem 59 - Algebra and Trigonometry

Question

A force is given by the vector \(\mathbf{F}=3 \mathbf{i}+2 \mathbf{j} .\) The force moves an object along a straight line from the point \((4,9)\) to the point \((10,20) .\) Find the work done if the distance is measured in feet and the force is measured in pounds.

Step-by-Step Solution

Answer

The work done is 40 foot-pounds.

Step 1: Identify Displacement Vector

To calculate the displacement vector, \( \mathbf{D} \), subtract the initial position vector from the final position vector. That is, \( \mathbf{D} = \mathbf{r_2} - \mathbf{r_1} \), where \( \mathbf{r_1} = 4\mathbf{i} + 9\mathbf{j} \) and \( \mathbf{r_2} = 10\mathbf{i} + 20\mathbf{j} \). We find \( \mathbf{D} = (10\mathbf{i} + 20\mathbf{j}) - (4\mathbf{i} + 9\mathbf{j}) = 6\mathbf{i} + 11\mathbf{j} \).

Step 2: Calculate the Dot Product

The work done by a force is given by the dot product of the force vector and the displacement vector. We can calculate the dot product using the following formula: \( \mathbf{F} \cdot \mathbf{D} = | \mathbf{F} | | \mathbf{D} | \cos\theta \). Therefore, \( Work = \mathbf{F} \cdot \mathbf{D} = (3 \mathbf{i} + 2 \mathbf{j}) \cdot (6\mathbf{i} + 11\mathbf{j}) = 3*6 + 2*11 = 18 + 22 = 40 \).

Step 3: Interpret the Result

The result we've obtained is the work done when moving the object from the point \((4,9)\) to \((10,20)\). 40 is the amount of work done, and the unit for this work will be foot-pound since our distance is measured in feet and our force is measured in pounds.