Chapter 14.2, Problem 5E

(a)

To determine

To find: The work done by a force in moving an object from $A\left(-2,1\right)$ to $B\left(3,3\right)$ .

Answer to Problem 5E

The work done by a force in moving an object from $A$ and B is $13$ units.

Explanation of Solution

Given information:

There is given a force in moving an object from $A\left(-2,1\right)$ to $B\left(3,3\right)$ .

The formula for calculating the work done by a force in moving an object from $A\left(-2,1\right)$ to $B\left(3,3\right)$ is as follows,

  $W=\overrightarrow{F}\cdot \overrightarrow{d}$

Here, $\overrightarrow{F}=3i-j$ and $\overrightarrow{d}=O\overrightarrow{B}-O\overrightarrow{A}$

Now calculate,

  $\begin{array}{c}\overrightarrow{d}=O\overrightarrow{B}-O\overrightarrow{A}\\ =\left(3i+3j\right)-\left(-2i+j\right)\\ =5i+2j\end{array}$

Substitute the value of $\overrightarrow{F}$ and $\overrightarrow{d}$ in the given formula ,

  $\begin{array}{c}W=\overrightarrow{F}\cdot \overrightarrow{d}\\ =\left(3i-j\right)\cdot \left(5i+2j\right)\\ =\left(3\times 5\right)+\left(-1\times 2\right)\\ =13\end{array}$

Now calculate the work done from $B\left(3,3\right)$ to $C\left(5,0\right)$ .

Here, $\overrightarrow{F}=3i-j$ and $\overrightarrow{d}=2i-3j$ .

Substitute in the given formula,

  $\begin{array}{c}W=\overrightarrow{F}\cdot \overrightarrow{d}\\ =\left(3i-j\right)\cdot \left(2i-3j\right)\\ =\left(3\times 2\right)+\left(-1\times -3\right)\\ =9\end{array}$

Therefore, the work done from $A$ to $B$ is $13$ units and from $B$ to $C$ is $9$ units.

(b)

To determine

To find: The work done by a force from $A\left(-2,1\right)$ to $B\left(5,0\right)$ .

Answer to Problem 5E

The work done by a force in moving an object from $A$ to $C$ is $22$ units.

Explanation of Solution

Given information:

There is given a force in moving an object from $A\left(-2,1\right)$ to $B\left(5,0\right)$ .

The formula for calculating the work done from $A\left(-2,1\right)$ to $C\left(5,0\right)$ is as follows,

  $W=\overrightarrow{F}\cdot \overrightarrow{d}$

Here, $\overrightarrow{F}=3i-j$ and $\overrightarrow{d}=O\overrightarrow{C}-O\overrightarrow{A}$

Now calculate,

  $\begin{array}{c}\overrightarrow{d}=O\overrightarrow{C}-O\overrightarrow{A}\\ =\left(5i+0j\right)-\left(-2i+j\right)\\ =7i-j\end{array}$

Substitute the value of $\overrightarrow{F}$ and $\overrightarrow{d}$ in the given formula,

  $\begin{array}{c}W=\overrightarrow{F}\cdot \overrightarrow{d}\\ =\left(3i-j\right)\cdot \left(7i-j\right)\\ =\left(3\times 7\right)+\left(-1\times -1\right)\\ =22\end{array}$

Therefore, the work done from $A$ to $C$ is $22$ units.