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College Physics by Openstax Chapter 2 Problem 53

There is a 250-m-high cliff at Half Dome in Yosemite National Park in California. Suppose a boulder breaks loose from the top of this cliff.
(a) How fast will it be going when it strikes the ground?
(b) Assuming a reaction time of 0.300 s, how long will a tourist at the bottom have to get out of the way after hearing the sound of the rock breaking loose (neglecting the height of the tourist, which would become negligible anyway if hit)? The speed of sound is 335 m/s on this day.

Solution:

Part A

Consider the Figure A. We shall consider two position points — position 1 at the top of the cliff, and position 2 on the ground.

Position 1 is 250 meters above the ground (y=250 m), and since this is the initial position, t=0, and the initial velocity is vy=0.

Position 2 is on the ground (y=0), and we do not know the time and velocity at this point.

For this part, we will solve for the value of vy at position 2. 

\begin{align*}
\left( v_{y_{2}} \right)^2 &  = \left( v_{y_{1}} \right)^2+2a \Delta y \\
\left( v_{y_{2}} \right)^2 &  = \left( 0\ \text{m/s} \right)^2+2\left( -9.81\ \text{m/s}^2 \right)\left( 0\ \text{m}-250\ \text{m} \right) \\
\left( v_{y_{2}} \right)^2 &  = 4905\ \text{m}^2/\text{s}^2 \\
v_{y_{2}} & = \pm \sqrt{4905\ \text{m}^2/\text{s}^2} \\
v_{y_{2}} & = \pm \ 70.0\ \text{m/s}\\
v_{y_{2}} & = -  70.0\ \text{m/s} \qquad {\color{gold}\left( \text{Answer} \right) }\\
\end{align*}
Figure A

Since the boulder is moving downward at this point, we shall consider the negative sign of the velocity as an indication of the downward direction of the velocity.

To answer the question, the boulder is going at about 70.0 m/s downward when it strikes the ground.

Part B

Let:
t_s be the time for the sound to travel from the top of the cliff to the tourist;
t be the total time elapsed before the tourist can react;
t_b be the time for the rock to travel from the top of the cliff to the ground
t_t be the amount of time the tourist has to get out of the way after hearing the sound of the rock breaking loose

Since we are given the speed of sound, we have

\begin{align*}

t_s & = \frac{\text{height of the cliff}}{\text{speed of sound}} \\
t_s & = \frac{250\ \text{m}}{335\ \text{m/s}}\\
t_s & =0.746 \ \text{s}

\end{align*}

So, the tourist can react after

\begin{align*}

t & =t_s + \text{reaction time} \\
t & = 0.746\ \text{s}+0.300\ \text{s} \\
t &= 1.046\ \text{s}

\end{align*}

Then, we can compute for the total time it takes for the rock to travel from top to the ground. 

\begin{align*}

t_b & =\frac{v_{y_2}-v_{y_1}}{a} \\
t_b & =\frac{-70\ \text{m/s}-0\ \text{m/s}}{-9.81\ \text{m/s}^2} \\
t_b & = 7.14\ \text{s}

\end{align*}

Therefore, the tourist still has the time to get out of the way. That is,

\begin{align*}

t_t & =7.14\ \text{s}-1.046\ \text{s} \\
t_t & = 6.09\ \text{s} \qquad {\color{gold}\left( \text{Answer} \right) } \\

\end{align*}

\therefore the tourist has about 6.09 seconds to get out of the way.

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College Physics by Openstax Chapter 2 Problem 52

An object is dropped from a height of 75.0 m above ground level. (a) Determine the distance traveled during the first second. (b) Determine the final velocity at which the object hits the ground. (c) Determine the distance traveled during the last second of motion before hitting the ground.

Solution:

Consider Figure 1. The object was dropped from a height of 75.0 m. At the start of motion, the velocity is zero, v_{oy}=0.

The object traveled for a period of time t for the whole 75.0 m distance to the ground.

Part A

We are solving for the distance traveled by the object for the first 1 second. So, we have

\begin{align*}
\Delta y & = v_{oy} t + \frac{1}{2}at^2\\
\Delta y & = \left( 0 \ \text{m/s} \right)\left( 1 \ \text{s} \right)+\frac{1}{2}\left( -9.81 \ \text{m/s}^2 \right)\left( 1 \ \text{s} \right)^2 \\
\Delta y & = 0 -4.905 \ \text{m} \\
\Delta y & = -4.91 \ \text{m} \\
|\Delta y| & =4.91 \ \text{m}
\end{align*}

The negative sign of ฮ”y indicates that the direction of the displacement is downward. Since we are looking for the scalar value of the distance, the answer is 4.91 m.

Part B

So we now consider the two positions of the object as shown in the figure to the right. The initial height of the object is 75.0 m above the ground, and the initial velocity is 0.

At the ground, we know that the position of the object is 0 m above the ground, but we do not know the time and velocity. Therefore, to determine the velocity of the object at this point, we proceed as follows:

\begin{align*}
\left( v_2 \right)^2 & =\left( v_1 \right)^2+2a \Delta y \\
\left( v_2 \right)^2 & =\left( v_1 \right)^2+2a \left( y_2-y_1 \right) \\
v_2 & = \pm \sqrt{\left( v_1 \right)^2+2a \left( y_2-y_1 \right)}\\
v_2 & = \pm \sqrt{\left( 0\ \text{m/s} \right)^2+2\left( -9.81\ \text{m/s}^2 \right)\left( 0 \ \text{m}-75\ \text{m} \right)} \\
v_2 & = \pm \ 38.4\ \text{m/s}\\
v_2 & =- 38.4\ \text{m/s}\\
\end{align*}

Since the object is directing downwards when it hit the ground, the velocity is negative.

Part C

First, we calculate the total time of the object’s motion from the beginning to the ground. 

\begin{align*}
\Delta y & =\bcancel{v_{oy}t}+ \frac{1}{2}at^2 \\
\Delta y & = \frac{1}{2}at^2 \\
0 \text{m}-75\ \text{m} & = \frac{1}{2}\left( -9.81\ \text{m/s}^2 \right)t^2 \\
t^2& =\frac{-75\ \text{m}}{-4.905 \text{m/s}^2}\\
t&=\sqrt{\frac{75\ \text{m}}{4.905 \text{m/s}^2}}\\
t&=3.91 \ \text{s}
\end{align*}

Second, determine the total distance traveled from 0 s to 2.91 s, leaving out the last 1 s of the motion. 

\begin{align*}
\Delta y & = v_{oy} t + \frac{1}{2}at^2\\
\Delta y & = \left( 0 \ \text{m/s} \right)\left( 1 \ \text{s} \right)+\frac{1}{2}\left( -9.81 \ \text{m/s}^2 \right)\left( 2.91 \ \text{s} \right)^2 \\
\Delta y & = 0 -41.5 \ \text{m} \\
\Delta y & = -41.5 \ \text{m} \\
|\Delta y| & =41.5 \ \text{m}
\end{align*}

Finally, subtract this distance from the total distance traveled to get the distance traveled in the last 1 second.

\begin{align*}
y_{_{\text{last 1 sec}}} & = 75.0 \ \text{m}-41.5 \ \text{m} \\
y_{_{\text{last 1 sec}}} & = 33.5\ \text{m}
\end{align*}
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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 ยตm)/(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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