Chapter 3, Problem 70PQ

(a)

To determine

The value of $m$.

Answer to Problem 70PQ

The value of $m$ is $3.0$.

Explanation of Solution

In problem it is given that $\overrightarrow{\text{F}}\text{=}3.3\widehat{\text{i}}+6.3\widehat{\text{j}}$ and $\overrightarrow{\text{A}}\text{=}1.1\widehat{\text{i}}+2.1\widehat{\text{j}}$.

Write the relation between $\overrightarrow{\text{F}}$ and $\overrightarrow{\text{A}}$.

  $\overrightarrow{\text{F}}\text{=}m\overrightarrow{\text{A}}$ (I)

Here, $\overrightarrow{\text{F}}$ is the force vector, $\overrightarrow{\text{A}}$ is a vector and $m$ is a scalar quantity.

Conclusion:

Substitute $3.3\widehat{\text{i}}+6.3\widehat{\text{j}}$ for $\overrightarrow{\text{F}}$ and $1.1\widehat{\text{i}}+2.1\widehat{\text{j}}$ for $\overrightarrow{\text{A}}$ in equation (I) to expand the equation.

$\begin{array}{c}\left(3.3\widehat{\text{i}}+6.3\widehat{\text{j}}\right)\text{=}m\left(1.1\widehat{\text{i}}+2.1\widehat{\text{j}}\right)\\ =\left(1.1m\widehat{\text{i}}+2.1m\widehat{\text{j}}\right)\end{array}$

Equate coefficient of $\widehat{\text{i}}$ in above equation to get $m$.

  $\begin{array}{c}3.3=1.1m\\ m=\frac{3.3}{1.1}\\ =3\end{array}$

Equate coefficient of $\widehat{\text{j}}$ in above equation to get $m$.

  $\begin{array}{c}6.3=2.1m\\ m=\frac{6.3}{2.1}\\ =3\end{array}$

Therefore, the value of $m$ is $3.0$.

(b)

To determine

To draw $\overrightarrow{\text{F}}$ and $\overrightarrow{\text{A}}$ on the same coordinate system.

Answer to Problem 70PQ

The following figure shows the vector $\overrightarrow{\text{F}}$ and $\overrightarrow{\text{A}}$.

Explanation of Solution

The following figure shows the vector $\overrightarrow{\text{F}}$ and $\overrightarrow{\text{A}}$.

(c)

To determine

The magnitude and direction of $\overrightarrow{\text{A}}$ and $\overrightarrow{\text{F}}$ .

Answer to Problem 70PQ

The magnitude of $\overrightarrow{\text{A}}$ is $2.4$ and magnitude of $\overrightarrow{\text{F}}$ is $7.1$ and both vectors are at $62°$ with respect to $x$ axis.

Explanation of Solution

Write the expression for the magnitude of $\overrightarrow{\text{A}}$.

  $\left|\overrightarrow{\text{A}}\right|=\sqrt{A_x^2+A_y^2}$ (II)

Here, $\left|\overrightarrow{\text{A}}\right|$ is the magnitude of $\overrightarrow{\text{A}}$, $A_x$ and $A_y$ are the $x$ and $y$ components of vector $\overrightarrow{\text{A}}$.

Write the expression for the magnitude of $\overrightarrow{\text{F}}$.

  $\left|\overrightarrow{F}\right|=\sqrt{F_x^2+F_y^2}$ (III)

Here, $\left|\overrightarrow{\text{F}}\right|$ is the magnitude of $\overrightarrow{\text{F}}$, $F_x$ and $F_y$ are the $x$ and $y$ components of vector $\overrightarrow{\text{F}}$.

Write the expression for the angle that $\overrightarrow{\text{F}}$ makes with $x$ axis.

  ${\theta }_F={\tan }^{-1}\left(\frac{F_y}{F_x}\right)$ (IV)

Here, ${\theta }_F$ is the angle that $\overrightarrow{\text{F}}$ makes with $x$ axis.

Write the expression for the angle that $\overrightarrow{\text{A}}$ makes with $x$ axis.

  ${\theta }_A={\tan }^{-1}\left(\frac{A_y}{A_x}\right)$ (V)

Here, ${\theta }_A$ is the angle that $\overrightarrow{\text{A}}$ makes with $x$ axis

Conclusion:

Substitute $3.3$ for $F_x$ and $6.3$ for $F_y$ in equation (III) to get $\left|\overrightarrow{\text{F}}\right|$.

  $\begin{array}{c}\left|\overrightarrow{\text{F}}\right|=\sqrt{{\left(3.3\right)}^2+{\left(6.3\right)}^2}\\ =7.1\end{array}$

Substitute $1.1$ for $A_x$ and $2.1$ for $A_y$ in equation (II) to get $\left|\overrightarrow{\text{A}}\right|$.

  $\begin{array}{c}\left|\overrightarrow{\text{A}}\right|=\sqrt{{\left(1.1\right)}^2+{\left(2.1\right)}^2}\\ =2.4\end{array}$

Substitute $3.3$ for $F_x$ and $6.3$ for $F_y$ in equation (IV) to get ${\theta }_F$.

  $\begin{array}{c}{\theta }_F={\tan }^{-1}\left(\frac{6.3}{3.3}\right)\\ =62°\end{array}$

Substitute $1.1$ for $A_x$ and $2.1$ for $A_y$ in equation (V) to get $\left|\overrightarrow{\text{A}}\right|$.

  $\begin{array}{c}{\theta }_A={\tan }^{-1}\left(\frac{2.1}{1.1}\right)\\ =62°\end{array}$

Therefore, the magnitude of $\overrightarrow{\text{A}}$ is $2.4$ and magnitude of $\overrightarrow{\text{F}}$ is $7.1$ and both vectors are at $62°$ with respect to $x$ axis.