Chapter 8, Problem 1ICA

To determine

Complete the table by using the equation of weight.

Explanation of Solution

Given data:

	Pound-mass	kilogram	newton	Pound-force
Abbreviation		kg
Example	0.33	0.15	1.47	0.33
(a)	10
(b)			700

Calculation:

The pound can be used as a unit of force or mass. To distinguish mass in terms of pounds and force in terms of pounds, the units of mass in terms of pounds is pound-mass which is abbreviated as ${\text{lb}}_{\text{m}}$.

Similarly, the units of force in terms of pounds is pound-force which is abbreviated as ${\text{lb}}_{\text{f}}$.

The SI unit of force is Newtons, which is denoted as N.

Case (a):

Consider the conversion factor from kilogram to pound-mass.

$1\text{kg}=2.205{\text{lb}}_{\text{m}}$

Rearrange the conversion as follows.

$1{\text{lb}}_{\text{m}}=\frac{1}{2.205}\text{kg}$

Convert $\text{10}{\text{lb}}_{\text{m}}$(pound-mass) to kilograms.

$\text{10}{\text{lb}}_{\text{m}}=\text{10}\left(\text{1}{\text{lb}}_{\text{m}}\right)$

Substitute $\frac{1}{2.205}\text{kg}$ for $1{\text{lb}}_{\text{m}}$ to obtain mass in terms of kilograms.

$\begin{array}{c}\text{10}{\text{lb}}_{\text{m}}=\text{10}\left(\frac{1}{2.205}\text{kg}\right)\\ =\text{10}\left(0.4535\text{kg}\right)\\ =4.535\text{kg}\end{array}$

Since the object is on earth, therefore acceleration due to gravity $\left(a\right)$ is $9.8\text{m}/{\text{s}}^{\text{2}}$.

Consider the expression for force $\left(F\right)$.

$F=ma$ (1)

Here,

$m$ is the mass.

$a$ is the acceleration due to gravity.

Substitute $4.535\text{kg}$ for $m$ and $9.8\text{m}/{\text{s}}^{\text{2}}$ for $a$ in equation (1) to obtain the value in terms of newtons.

$\begin{array}{c}F=\left(4.535\text{kg}\right)\left(9.8\text{m}/{\text{s}}^{\text{2}}\right)\\ =44.443\text{N}\\ \cong 44.5\text{N}\end{array}$

Consider the conversion factor from Newtons to pound-force.

$1\text{N}=0.225{\text{lb}}_{\text{f}}$

Convert $\text{44}\text{.5}\text{N}$ to pound-force.

$\text{44}\text{.5}\text{N}=\text{44}\text{.5}\left(\text{1}\text{N}\right)$

Substitute $0.225{\text{lb}}_{\text{f}}$ for $1\text{N}$ to obtain the mass in terms of pound-force,

$\begin{array}{c}\text{44}\text{.5}\text{N}=\text{44}\text{.5}\left(0.225{\text{lb}}_{\text{f}}\right)\\ =10.0125{\text{lb}}_{\text{f}}\\ \cong 10{\text{lb}}_{\text{f}}\end{array}$

Case (b):

Convert $\text{700}\text{N}$ to pound-force.

$\text{700}\text{N}=\text{700}\left(\text{1}\text{N}\right)$

Substitute $0.225{\text{lb}}_{\text{f}}$ for $1\text{N}$ to obtain the mass in terms of pound-force,

$\begin{array}{c}\text{700}\text{N}=\text{700}\left(0.225{\text{lb}}_{\text{f}}\right)\\ =157.5{\text{lb}}_{\text{f}}\end{array}$

Rearrange equation (1) to obtain the mass in terms of kilograms.

$m=\frac{F}{a}$

Substitute 700 N for F and $9.8\text{m}/{\text{s}}^{\text{2}}$ for $a$ to obtain the mass in terms of kilograms.

$\begin{array}{c}m=\frac{700\text{N}}{9.8\text{m}/{\text{s}}^{\text{2}}}\\ =\frac{700\text{N}\cdot {\text{s}}^{\text{2}}}{9.8\text{m}}\\ =71.42\text{kg}\left\{\because 1\text{kg}=\frac{\text{N}\cdot {\text{s}}^{\text{2}}}{\text{m}}\right\}\end{array}$

Convert $\text{71}\text{.42}\text{kg}$ to pound-mass.

$\text{71}\text{.42}\text{kg}=71.42\left(\text{1}\text{kg}\right)$

Substitute $2.205{\text{lb}}_{\text{m}}$ for $1\text{kg}$ to obtain the mass in terms of pound-mass,

$\begin{array}{c}\text{71}\text{.42}\text{kg}=\text{71}\text{.42}\left(2.205{\text{lb}}_{\text{m}}\right)\\ =157.48{\text{lb}}_{\text{m}}\\ \cong 157.5{\text{lb}}_{\text{m}}\end{array}$

Complete the table by using the calculated values as shown in Table 1.

Table 1

	Pound-mass	kilogram	newton	Pound-force
Abbreviation	${\text{lb}}_{\text{m}}$	kg	N	${\text{lb}}_{\text{f}}$
Example	0.33	0.15	1.47	0.33
(a)	10	4.535	44.5	10
(b)	157.5	71.42	700	157.5

Conclusion:

Thus, the unknown parameters in the table are calculated.