Author Archives: Engineering Math

Hibbeler Statics 14E P2.4 – Components of a Force Along Two Non-Perpendicular Axes

The vertical force \textbf{F} acts downward at A on the two-membered frame. Determine the magnitudes of the two components of \textbf{F} directed along the axes of AB and AC. Set \textbf{F} = 500 N.

Engineering Mechanics: Statics 14th Edition by RC Hibbeler, Problem 2-4

Solution:

Draw the components of the force using the parallelogram law. Then the triangulation rule.

Parallelogram Law
Triangulation Rule

Solving for FAC using sine law.

\begin{align*}
\frac{\text{F}_\text{AC}}{\sin \ 45^\circ } & =\frac{500 \ \text{N}}{\sin \ 75^\circ }\\
\text{F}_\text{AC} & = \frac{500 \ \text{N} \ \sin45^\circ }{\sin\ 75^\circ }\\
\text{F}_\text{AC} & =366.0254 \ \text{N}\\
\text{F}_\text{AC} &\approx 366 \ \text{N}
\end{align*}

Solve for FAB using sine law.

\begin{align*}
\frac{\text{F}_\text{AB}}{\sin \ 60^\circ } & =\frac{500 \ \text{N}}{\sin \ 75^\circ }\\
\text{F}_\text{AB} & = \frac{500 \ \text{N} \ \sin60^\circ }{\sin\ 75^\circ }\\
\text{F}_\text{AB} & =448.2877 \ \text{N}\\
\text{F}_\text{AB} &\approx 448 \ \text{N}
\end{align*}
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Hibbeler Statics 14E P2.2 — Solving for an Unknown Force Given the Magnitude and Direction of a Resultant and Another Force

If the magnitude of the resultant force is to be 500 N, directed along the positive y-axis, determine the magnitude of force F and its direction \theta .

Engineering Mechanics: Statics 14th Edition by RC Hibbeler, Problem 2-2

Solution:

The parallelogram law and the triangulation rule are shown in the figures below.

Engineering Mechanics by RC Hibbeler Problem 2.2 Parallelogram Law
Parallelogram Law
Engineering Mechanics by RC Hibbeler Problem 2.2 Triangulation Rule
Triangulation Rule

Considering the figure of the triangulation rule, we can solve for the magnitude of \textbf{F} using the cosine law.

\begin{align*}
\textbf{F} & = \sqrt{700^2+500^2-2\left( 700 \right)\left( 500 \right)\cos105^{\circ}}\\
& = 959.78 \  \text{N}\\
& = 960 \  \text{N}\\
\end{align*}

Then we use the sine law to solve for the angle \theta. 

\begin{align*}
\frac{\sin \left(90^{\circ}-\theta \right)}{700} & = \frac{\sin 105^{\circ}}{959.78}\\
\sin \left(90^{\circ}-\theta \right) & =\frac{700 \sin 105^{\circ }}{959.78}\\
90^{\circ}-\theta & = \sin^{-1} \left( \frac{700 \sin 105^{\circ }}{959.78} \right)\\
\theta & = 90^\circ-\sin^{-1} \left( \frac{700 \sin 105^{\circ }}{959.78} \right) \\
\theta & =  90^\circ-44.79^\circ\\
\theta & =  45.2^\circ\\
\end{align*}
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Hibbeler Statics 14E P2.1 — Solving for the Magnitude and Direction of the Resultant of Two Coplanar-Concurrent Forces

If \theta = 60 \degree and \textbf{F} = 450 \ \text{N}, determine the magnitude of the resultant force and its direction, measured counterclockwise from the positive x axis.

Engineering Mechanics: Statics 14th Edition by RC Hibbeler, Problem 2-1

Solution:

The parallelogram law and the triangulation rule are shown in the figures below.

(a) Parallelogram Law
(b) Triangulation Rule

Considering figure (b), we can solve for the magnitude of \textbf{F}_R using the cosine law.

\begin{align*}
\textbf{F}_R & = \sqrt{700^2+450^2-2\left( 700 \right)\left( 450 \right)\cos45^{\circ}}\\
& = 497.01 \ \text{N}\\
& = 497 \ \text{N}
\end{align*}

Then we use the sine law to solve for the interior angle \theta. 

\begin{align*}
\frac{\sin \theta}{700} & = \frac{\sin 45^{\circ}}{497.01}\\
\sin \theta & =\frac{700\ \sin 45^{\circ }}{497.01}\\
\theta & = \sin^{-1} \left( \frac{700\ \sin 45^{\circ }}{497.01} \right)\\
& \text{This is an ambiguous case }\\
\theta & = 84.81^\circ \  or \  \theta =95.19^\circ \\
\end{align*}

In here, the correct angle measurement is \theta = 95.19^{\circ}.

Thus, the direction angle \phi of \textbf{F}_R measured counterclockwise from the positive x-axis, is 

\begin{align*}
\phi & = \theta +60^\circ \\
& = 95.19^\circ +60^\circ \\
& = 155^\circ 
\end{align*}
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Hibbeler Statics 14E P1.19 — Determine the Weight of the Column with a given Density

A concrete column has a diameter of 350 mm and a length of 2 m. If the density (mass/volume) of concrete is 2.45 Mg/m3, determine the weight of the column in pounds.

Engineering Mechanics: Statics 14th Edition by RC Hibbeler, Problem 1-19

Solution:

The density of any material is given by the formula

\text{density}=\frac{\text{mass}}{\text{volume}}

From there, we can compute for the mass as

\text{mass}=\text{density} \times \text{volume}

We can solve for mass by multiplying density by volume. The density is already given, and we can compute for the volume of the concrete column by the formula of a volume of a cylinder.

\begin{align*}
\text{V} & = \pi \text{r}^2 \text{h}\\
& =\pi \left( \frac{0.35\ \text{m}}{2} \right)^2 \left( 2 \ \text{m} \right)\\
& =0.1924 \ \text{m}^3
\end{align*}
Concrete Column illustration with diameter of 350 mm or 0.35 m, and a height of 2 m

Therefore, the mass of the concrete column is 

\begin{align*}
\text{mass} & =\text{density} \times \text{volume}\\
& = \left( 2.45 \times 10^3 \ \text{kg/m}^3 \right)\times \left( 0.1924 \ \text{m}^3 \right)\\
& =471.44 \ \text{kg}\\
\end{align*}

Now, we can solve for the weight by multiplying the mass by the acceleration due to gravity, g.

\begin{align*}
\text{Weight} & = \text{mass} \times \text{acceleration due to gravity} \\
& = 471.44 \ \text{kg} \times 9.81 \ \text{m/s}^2 \\
& = 4624.78 \ \text{N}
\end{align*}

Finally, we can convert the weight in Newtons to weight in pounds. 

\begin{align*}
4624.78\ \text{N} & = 4624.78\ \text{N}\times \frac{1\ \text{lb}}{4.4482\ \text{N}}\\
& = 1039.70\ \text{lb}\\
& = 1.04\times 10^3 \  \text{lb}\\
& = 1.04 \ \text{kip}
\end{align*}
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Hibbeler Statics 14E P1.14 — Evaluation of expression to correct SI Units

Evaluate each of the following to three significant figures and express each answer in SI units using an appropriate prefix: (a) (212 mN)2, (b) (52800 ms)2, and (c) [548(106)]1/2 ms.

Engineering Mechanics: Statics 14th Edition by RC Hibbeler, Problem 1-14

Solution:

Part A

\begin{align*}
\left( 212 \ \text{mN} \right)^2 & = \left[ 212\times 10^{-3} \ \text{N} \right]^2 \\
& = 0.0449 \ \text{N}^2 \\
& = 4.49\times 10^{-2} \ \text{N}^2\\
\end{align*}

Part B

\begin{align*}
\left( 52800 \ \text{ms} \right)^2 & = \left[ 52800\times 10^{-3} \ \text{s} \right]^2 \\
& =2788 \ \text{s}^2 \\
& = 2.79 \times 10^3 \ \text{s}^2
\end{align*}

Part C

\begin{align*}
\left[ 548\left( 10^6 \right) \right]^{1/2} \ \text{ms}& =23409 \ \text{ms}\\
& =23409\times 10^{-3}\ \text{s}\\
& = 23.4\times 10^3\times 10^{-3} \ \text{s}\\
& = 23.4 \ \text{s}
\end{align*}
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Hibbeler Statics 14E P1.13 — Conversion of Density from Slug per Cubic Foot to Appropriate SI Unit

The density (mass volume) of aluminum is 5.26 slug/ft3. Determine its density in SI units. Use an appropriate prefix.

Engineering Mechanics: Statics 14th Edition by RC Hibbeler, Problem 1-13

Solution:

\begin{align*}
5.26 \ \text{slug/ft}^3 & =\left( \frac{5.26 \ \text{slug}}{\text{ft}^3} \right)\left( \frac{1 \ \text{ft}}{0.3048\ \text{m}} \right)^3\left( \frac{14.59\ \text{kg}}{1\ \text{slug}} \right)\\
& = 2710\ \text{kg/m}^3\\
& = 2.71\times 10^3 \ \text{kg/m}^3\\
& = 2.71\ \text{Mg/m}^3
\end{align*}
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Hibbeler Statics 14E P1.12 — Evaluation of Expression to Three Significant Figures with Appropriate SI Units

Evaluate each of the following to three significant figures and express each answer in SI units using an appropriate prefix: (a) (684 µm)/(43 ms), (b) (28 ms)(0.0458 Mm)/(348 mg), (c) (2.68 mm)(426 Mg).

Statics of Rigid Bodies 14th Edition by RC Hibbeler, Problem 1-12

Solution:

Part A

\begin{align*}
\left( 684 \ \mu\text{m} \right)/43 \ \text{ms} & =\frac{684\times 10^{-6} \ \text{m}}{43\times 10^{-3} \ \text{s}}\\
& = \frac{15.9\times 10^{-3}\ \text{m}}{\text{s}}\\
& = 15.9 \  \text{mm/s}
\end{align*}

Part B

\begin{align*}
\left( 28 \  \text{ms} \right)\left( 0.0458 \ \text{Mm} \right)/\left( 348 \ \text{mg} \right) & = \frac{\left[ 28\times 10^{-3} \ \text{s} \right]\left[ 45.8\times 10^{-3}\times 10^6 \ \text{m} \right]}{348\times 10^{-3}\times 10^{-3} \ \text{kg}} \\
& = \frac{3.69\times 10^6 \ \text{m}\cdot \text{s}}{\text{kg}}\\
& = 3.69 \  \text{Mm}\cdot \text{s}/\text{kg}\\
\end{align*}

Part C

\begin{align*}
\left( 2.68 \ \text{mm} \right)\left( 426 \ \text{Mg} \right) & = \left[ 2.68\times 10^{-3} \ \text{m} \right]\left[ 426\times 10^3 \ \text{kg} \right]\\
& = 1.14\times 10^3 \ \text{m}\cdot \text{kg}\\
& = 1.14 \  \text{km}\cdot \text{kg}\\
\end{align*}
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Hibbeler Statics 14E P1.11 — Representing Measurements with SI units Having Appropriate Prefix

Represent each of the following with SI units having an appropriate prefix: (a) 8653 ms, (b) 8368 N, (c) 0.893 kg.

Statics of Rigid Bodies 14th Edition by RC Hibbeler, Problem 1-11

Solution:

Part A

\begin{align*}
8653 \ \text{ms} & = 8653 \ \left( 10^{-3} \right) \ \text{s} \\
& =8.653 \ \text{s}
\end{align*}

Part B

\begin{align*}
8368 \ \text{N} & = 8.368\times 10^3 \ \text{N}\\
& = 8.368 \ \text{kN}\\
\end{align*}

Part C

\begin{align*}
0.893 \  \text{kg} & = 0.893\times 10^3 \ \text{g} \\
& =893 \ \text{g}
\end{align*}
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Determining if a given Differential Equation is Separable or Not

Determine whether each of the following differential equations is or is not separable, and, if it is separable, rewrite the equation in the form dy/dx=f(x) g(y).
\qquad \textbf{a}) \quad \frac{dy}{dx}=xy-3x-2y+6
\qquad \textbf{b})\quad \frac{dy}{dx}=\sin \left( x+y \right)
\qquad \textbf{c}) \quad y\frac{dy}{dx}=e^{x-3y^2}

Solution:

Part A

\begin{align*}
\frac{dy}{dx} & = xy-3x-2y+6 \\
\frac{dy}{dx} & = \left( xy-3x \right)-\left( 2y-6 \right)\\
\frac{dy}{dx} & = x\left( y-3 \right)-2\left( y-3 \right)\\
\frac{dy}{dx} & = \left( x-2 \right)\left( y-3 \right)
\end{align*}

Since F(x,y) is factorable in the form f(x) g(y), the given differential equation is separable.

Part B

\begin{align*}
\frac{dy}{dx} & = \sin\left( x+y \right) \\
\frac{dy}{dx} & = \sin\left( x \right)\cos\left( y \right) +\cos\left( x \right)\sin\left( y \right)\\
\end{align*}

Since F(x,y) is not factorable in the form f(x) g(y), the given differential equation is not separable.

Part C

\begin{align*}
y \frac{dy}{dx} & = e^{x-3y^2}\\
y \frac{dy}{dx} & = \frac{e^x}{e^{3y^2}}\\
\frac{dy}{dx} & =\frac{e^x}{y e^{3y^2}} \\
\frac{dy}{dx} & = e^x \left( \frac{1}{ye^{3y^2}} \right) \\
\end{align*}

Since F(x,y) is factorable in the form f(x) g(y), the given differential equation is separable.

Hibbeler Statics 14E P1.10 — Representing Combinations of Units in the Correct SI Form

Represent each of the following combinations of units in the correct SI form: (a) GN∙µm, (b) kg/µm, (c) N/ks2, and (d) KN/µs.

Statics of Rigid Bodies 14th Edition by RC Hibbeler, Problem 1-10

Solution:

Part A

\begin{align*}
\text{GN} \cdot  \mu \text{m} & = \left( 10^9 \ \text{N} \right)\left( 10^{-6} \ \text{m} \right)\\
& = 10^3 \ \text{N} \cdot \text{m}\\
& = \text{kN} \cdot \text{m}
\end{align*}

Part B

\begin{align*}
\text{kg/}\mu\text{m} & = \frac{10^3 \ \text{g}}{10^{-6} \ \text{m}} \\
& = 10^9 \ \frac{\text{g}}{\text{m}} \\
& = \text{Gg/m}
\end{align*}

Part C

\begin{align*}
\text{N/ks}^2 & = \frac{\text{N}}{\left( 10^3 \ \text{s} \right)^2}\\
& = \frac{\text{N}}{10^6 \ \text{s}^2} \\
& = 10^{-6} \ \frac{\text{N}}{\text{s}^2} \\
& = \mu \text{N}/\text{s}^2
\end{align*}

Part D

\begin{align*}
\text{kN}/ \mu\text{s} & = \frac{10^3 \ \text{N}}{10^{-6} \ \text{s}} \\
& = 10^9 \ \frac{\text{N}}{\text{s}}\\
& = \text{GN/s}
\end{align*}
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