Tag Archives: Concept of Statics

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

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

FACsin⁡ 45∘=500 Nsin⁡ 75∘FAC=500 N sin⁡45∘sin⁡ 75∘FAC=366.0254 NFAC≈366 N\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.

FABsin⁡ 60∘=500 Nsin⁡ 75∘FAB=500 N sin⁡60∘sin⁡ 75∘FAB=448.2877 NFAB≈448 N\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 θ \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 F \textbf{F} using the cosine law.

F=7002+5002−2(700)(500)cos⁡105∘=959.78 N=960 N\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 θ\theta.

sin⁡(90∘−θ)700=sin⁡105∘959.78sin⁡(90∘−θ)=700sin⁡105∘959.7890∘−θ=sin⁡−1(700sin⁡105∘959.78)θ=90∘−sin⁡−1(700sin⁡105∘959.78)θ=90∘−44.79∘θ=45.2∘\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 θ=60°\theta = 60 \degree and F=450 N \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 FR \textbf{F}_R using the cosine law.

FR=7002+4502−2(700)(450)cos⁡45∘=497.01 N=497 N\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 θ\theta.

sin⁡θ700=sin⁡45∘497.01sin⁡θ=700 sin⁡45∘497.01θ=sin⁡−1(700 sin⁡45∘497.01)This is an ambiguous case θ=84.81∘ or θ=95.19∘\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 θ=95.19∘ \theta = 95.19^{\circ}.

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

ϕ=θ+60∘=95.19∘+60∘=155∘\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

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

From there, we can compute for the mass as

mass=density×volume\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.

V=πr2h=π(0.35 m2)2(2 m)=0.1924 m3\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

mass=density×volume=(2.45×103 kg/m3)×(0.1924 m3)=471.44 kg\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.

Weight=mass×acceleration due to gravity=471.44 kg×9.81 m/s2=4624.78 N\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.

4624.78 N=4624.78 N×1 lb4.4482 N=1039.70 lb=1.04×103 lb=1.04 kip\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

(212 mN)2=[212×10−3 N]2=0.0449 N2=4.49×10−2 N2\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

(52800 ms)2=[52800×10−3 s]2=2788 s2=2.79×103 s2\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

[548(106)]1/2 ms=23409 ms=23409×10−3 s=23.4×103×10−3 s=23.4 s\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:

5.26 slug/ft3=(5.26 slugft3)(1 ft0.3048 m)3(14.59 kg1 slug)=2710 kg/m3=2.71×103 kg/m3=2.71 Mg/m3\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

(684 μm)/43 ms=684×10−6 m43×10−3 s=15.9×10−3 ms=15.9 mm/s\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

(28 ms)(0.0458 Mm)/(348 mg)=[28×10−3 s][45.8×10−3×106 m]348×10−3×10−3 kg=3.69×106 m⋅skg=3.69 Mm⋅s/kg\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

(2.68 mm)(426 Mg)=[2.68×10−3 m][426×103 kg]=1.14×103 m⋅kg=1.14 km⋅kg\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 P2.2 — Resultant of a System of Two Forces

Determine the magnitude of the resultant force FR=F1+F2\textbf{F}_{\text{R}} = \textbf{F}_1 + \textbf{F}_2 and its direction, measured counterclockwise from the positive x axis. 

Engineering Mechanics: Statics figure for Problem 2-3

Engineering Mechanics: Statics 13th Edition by RC Hibbeler, Problem 2-1
Engineering Mechanics: Statics 14th Edition by RC Hibbeler, Problem 2-3

SOLUTION:

The parallelogram law of the force system is shown.

Consider the triangle AOB.

Using cosine law to solve for the resultant force FR\textbf{F}_{\text{R}}

FR=(250)2+(375)2−2(250)(375)cos⁡ 75∘=393.2 lb=393 lb\begin{align*} \textbf{F}_\text{R} & =\sqrt{\left(250\right)^2+\left(375\right)^2-2\left(250\right)\left(375\right) \cos\:75^{\circ} }\\ & =393.2 \ \text{lb}\\ & =393\:\text{lb}\\ \end{align*}

The value of angle θ can be solved using sine law. 

393.2sin⁡ (75∘)=250sin⁡ θsin⁡θ=250 sin⁡75°393.2θ=sin⁡−1(250 sin⁡75°393.2)θ=37.89∘\begin{align*} \frac{393.2}{\sin\:\left(75^{\circ} \right)} & = \frac{250}{\sin\:\theta } \\ \sin \theta & = \frac{250 \ \sin75 \degree}{393.2}\\ \theta & =\sin^{-1} \left(\frac{250 \ \sin75 \degree}{393.2}\right)\\ \theta & = 37.89^{\circ}\\ \end{align*}

Solve for the unknown angle ϕ \phi .

ϕ=360∘−45∘+37.89∘=353∘\phi =360^{\circ} -45^{\circ} +37.89^{\circ} =353^{\circ}

The resultant force has a magnitude of 393 lb and is located 353º measured counterclockwise from the positive x-axis.

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Computing the mass and weight of a man on earth and on the moon

If a man weighs 155 lb on earth, specify (a) his mass in slugs, (b) his mass in kilograms, and (c) his weight in newtons. If the man is on the moon, where the acceleration due to gravity is gm=5.30 ft/s², determine (d) his weight in pounds, and (e) his mass in kilograms.

Engineering Mechanics: Statics 13th Edition by RC Hibbeler, Problem 1-21
Engineering Mechanics: Statics 14th Edition by RC Hibbeler, Problem 1-20

Solution:

Part A

From the formula, W=mg\text{W}=\text{mg}, we can solve for the mass by dividing the weight by the acceleration due to gravity. That is

m=Wg=155 lb32.2 ft/s2=4.81 slug\begin{align*} \text{m} & = \frac{\text{W}}{\textbf{g}}\\ & = \frac{155\ \text{lb}}{32.2 \ \text{ft/s}^2}\\ & = 4.81 \ \text{slug}\\ \end{align*}

Part B

Convert the slug to kilograms, knowing that 1 slug = 14.59 kg.

m=(15532.2slug)(14.59 kg1 kg)=70.2 kg\begin{align*} \begin{align*} \text{m} & = \left( \frac{155}{32.2} \text{slug}\right)\left( \frac{14.59 \ \text{kg}}{1 \ \text{kg}} \right)\\ & = 70.2 \ \text{kg}\\ \end{align*} \end{align*}

Part C

Convert the 155 lb to newtons using 1 lb = 4.448 N.

W=155 lb×4.448 N1 lb=689 N\begin{align*} \textbf{W} & = 155 \ \text{lb}\times \frac{4.448 \ \text{N}}{1 \ \text{lb}}\\ & = 689 \ \text{N}\\ \end{align*}

Part D

Using the same formulas, but now g=5.30 ft/s2\textbf{g}=5.30 \ \text{ft/s}^2.

W=155(5.3032.2)=25.5 lb\textbf{W}=155\left(\frac{5.30}{32.2}\right)=25.5\:\text{lb}

Part E

m=155(14.59 kg32.2)=70.2 kg\textbf{m}=155\left(\frac{14.59\:\text{kg}}{32.2}\right)=70.2\:\text{kg}
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The force of gravity acting between two particles

Two particles have a mass of 8 kg and 12 kg, respectively. If they are 800 mm apart, determine the force of gravity acting between them. Compare this result with the weight of each particle.

Engineering Mechanics: Statics 13th Edition by RC Hibbeler, Problem 1-20
Engineering Mechanics: Statics 14th Edition by RC Hibbeler, Problem 1-21

Solution:

The force of gravity acting between them:

F=G⋅m1m2r2=66.73(10−12)m3/(kg⋅s2)[8 kg(12 kg)(0.8 m)2]=10(10−9) N=10.0 nN\begin{align*} \textbf{F} & =\textbf{G}\cdot \frac{\text{m}_1\text{m}_2}{\text{r}^2}\\ & =66.73\left(10^{-12}\right) \text{m}^3/ \left( \text{kg} \cdot \text{s}^2 \right) \left[\frac{8 \ \text{kg} \left(12\ \text{kg}\right)}{\left(0.8\ \text{m} \right)^2}\right]\\ &=10\left(10^{-9}\right)\ \text{N}\\ & =10.0 \ \text{nN}\\ \end{align*}

The weight of the 8 kg particle

W1=8(9.81)=78.5 N\textbf{W}_1=8\left(9.81\right)=78.5\:\text{N}

Weight of the 12 kg particle

W2=12(9.81)=118 N\textbf{W}_2=12\left(9.81\right)=118\:\text{N}
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