Tag Archives: Physics PDF solutions

Problem 6-20: The centripetal acceleration of the commercial jet’s tires, and the force of a determined bacterium in it


At takeoff, a commercial jet has a 60.0 m/s speed. Its tires have a diameter of 0.850 m.

(a) At how many rev/min are the tires rotating?

(b) What is the centripetal acceleration at the edge of the tire?

(c) With what force must a determined 1.00×10−15 kg bacterium cling to the rim?

(d) Take the ratio of this force to the bacterium’s weight.


Solution:

We are given the following quantities: linear speed, v=60.0 \ \text{m/s}, radius is half the diameter, r=0.425 \ \text{m}.

Part A

We can compute the angular velocity based on the given using the formula, \displaystyle \omega = \frac{v}{r}.

\begin{align*}
\omega & = \frac{v}{r} \\ \\
\omega & = \frac{60.0 \ \text{m/s}}{0.425 \ \text{m}} \\ \\
\omega & = 141.1765 \ \text{rad/sec}
\end{align*}

Now, we can convert this into the required unit of rev/min.

\begin{align*}
\omega & = 141.1765\ \frac{\text{rad}}{\text{sec}} \times \frac{1\ \text{rev}}{2\pi\ \text{rad}} \times \frac{60\ \text{sec}}{1\ \text{min}} \\ \\
\omega & = 1348.1363 \ \text{rev/min} \\ \\
\omega & = 1.35 \times 10^{3} \ \text{rev/min} \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right)
\end{align*}

Part B

The centripetal acceleration at the edge of the tire can be computed using the formula, a_{c} = r \omega ^{2}.

\begin{align*}
a_{c} & = r \omega ^2 \\ \\
a_{c} & = \left( 0.425\ \text{m} \right) \left(141.1765\ \text{rad/sec} \right)^2 \\ \\
a_{c} & = 8470.5918 \ \text{m/s}^2 \\ \\ 
a_{c} & = 8.47 \times 10 ^{3} \ \text{m/s}^2 \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right)
\end{align*}

Part C

From the second law of motion, the force is equal to the product of the mass and the acceleration. In this case, we are going to use the formula, F_c = m a_c . We are given the mass to be m=1.00 \times 10 ^{-15}\ \text{kg} , and the centripetal acceleration is solved in Part B.

\begin{align*}
F_c & = ma_c \\ \\
F_c & = \left( 1 \times 10^{-15}\ \text{kg}\right) \left(8470.5918 \ \text{m/s}^2\right) \\ \\
F_c & = 8.4705918 \times 10 ^{-12}\ \text{kg m/s}^2 \\ \\
F_c & = 8.47 \times 10^{-12} \ \text{N} \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right)
\end{align*}

Part D

The ratio of this force, F_c to the weight of the bacterium is

\begin{align*}
\frac{F_c}{mg} & = \frac{8.4705819 \times 10 ^{-12}\ \text{N}}{\left( 1 \times 10^{-15} \text{kg}  \right)\left(9.81 \ \text{m/s}^2 \right)} \\ \\
\frac{F_c}{mg} & = 863.4640 \\ \\
\frac{F_c}{mg} & = 863 \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right)
\end{align*}

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Problem 6-18: The linear speed of an ultracentrifuge and Earth in its orbit


Verify that the linear speed of an ultracentrifuge is about 0.50 km/s, and Earth in its orbit is about 30 km/s by calculating:

(a) The linear speed of a point on an ultracentrifuge 0.100 m from its center, rotating at 50,000 rev/min.

(b) The linear speed of Earth in its orbit about the Sun (use data from the text on the radius of Earth’s orbit and approximate it as being circular).


Solution:

Part A

We are given a linear speed of an ultracentrifuge of 0.50\ \text{km/s}. We are asked to verify this value if we are given a radius of r=0.100\ \text{m} and angular velocity of \omega = 50000 \ \text{rev/min}. We are going to use the formula

v = r \omega

Since we are given a linear speed in \text{km/s}, we are going to convert the radius to \text{km}, and the angular velocity to \text{rad/sec}

r=0.100\ \text{m} \times \frac{1\ \text{km}}{1000\ \text{m}} = 0.0001\ \text{km} 
\omega = 50000 \ \text{rev/min} \times \frac{2\pi \ \text{rad}}{1\ \text{rev}} \times \frac{1\ \text{min}}{60\ \text{sec}} =5235.9878\ \text{rad/sec}

Now, we can substitute these into the formula

\begin{align*}
v & = r \omega \\ \\
v & = \left( 0.0001 \ \text{km} \right)\left( 5235.9878 \ \text{rad/sec} \right) \\ \\
v & = 0.5236 \ \text{km/s} \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right)
\end{align*}

This value is about 0.500 km/s.

Part B

From Table 6.2 of the book

ParentSatelliteAverage orbital radius r(km)Period T(y)r3 / T2 (km3 / y2)
SunEarth1.496 \times 10^{8} 13.35 \times 10^{24}

Using the same formulas we used in Part A, we can solve for the linear velocity of the Earth around the sun. The radius is

r=1.496 \times10^{8} \ \text{km}

The angular velocity is

\begin{align*}
\omega &  = 1 \ \frac{\text{rev}}{\text{year}} \times \frac{2\pi \ \text{rad}}{1\ \text{rev}} \times \frac{1 \ \text{year}}{365.25 \ \text{days}} \times \frac{1\ \text{day}}{24\ \text{hours}}\times \frac{1\ \text{hour}}{3600\ \text{sec}} \\ \\ 
\omega &  = 1.9910 \times 10^{-7}\ \text{rad/sec}
\end{align*}

The linear velocity is

\begin{align*}
v & = r \omega \\ \\
v & = \left( 1.496\times 10^{8}\ \text{km} \right)\left( 1.9910 \times 10 ^ {-7} \right) \ \text{rad/sec}\\ \\
v & = 29.7854\ \text{km/s}\ \qquad \ \color{DarkOrange} \left( \text{Answer} \right)
\end{align*}

The linear velocity is about 30 km/s.


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Problem 6-17: The acceleration due to gravity at the position of a satellite located above the Earth


What percentage of the acceleration at Earth’s surface is the acceleration due to gravity at the position of a satellite located 300 km above Earth?


Solution:

The acceleration due to gravity of a body and the Earth is given by the formula

g= G \frac{M}{r^2}

where G is the gravitational constant, M is the mass of the Earth, and r is the distance of the object to the center of the Earth. We know that the approximate radius of the Earth is r=6.3781 \times 10^6 \ \text{m} .

The percentage of the acceleration at 300 km above the Earth of the acceleration due to gravity at Earth’s surface is

\displaystyle \frac{\left( \frac{GM}{r^2} \right)_2}{\left( \frac{GM}{r^2} \right)_1} \times 100\%

Note that the subscript 2 indicates the satellite located 300 km above the Earth, and the subscript 1 indicates the object at the Earth’s surface. Also, from the expression above, we can cancel G and M from the numerator and denominator because these are constants. So, we are down to

\frac{\left( \frac{1}{r^2} \right)_2}{\left( \frac{1}{r^2} \right)_1} \times 100\% = \frac{\left( r^2 \right)_1}{\left( r^2 \right)_2} \times 100\%

Substituting the values, we have

\begin{align*}
\frac{\left( r^2 \right)_1}{\left( r^2 \right)_2} \times 100\% & = \frac{\left( 6.3781 \times 10^6 \ \text{m} \right)^{2}}{\left( 6.3781 \times 10^6 \ \text{m}+300 \times 10^{3} \ \text{m} \right)^{2}} \times 100\% \\ \\
& = 91.2172\% \\ \\
& = 91.2\% \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right)
\end{align*}

The percentage of the acceleration at the Earth’s surface of the acceleration due to gravity at the position of a satellite located 300 km above the Earth is about 91.2%.


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Problem 6-16: Calculating the centripetal acceleration of an ice skater’s nose


Olympic ice skaters are able to spin at about 5.00 rev/s.

(a) What is their angular velocity in radians per second?

(b) What is the centripetal acceleration of the skater’s nose if it is 0.120 m from the axis of rotation?

(c) An exceptional skater named Dick Button was able to spin much faster in the 1950s than anyone since—at about 9.00 rev/s. What was the centripetal acceleration of the tip of his nose, assuming it is at 0.120 m radius?

(d) Comment on the magnitudes of the accelerations found. It is reputed that Button ruptured small blood vessels during his spins.


Solution:

We are given an angular velocity, \omega = 5 \ \text{rev/sec}

Part A

For this part, we are asked to convert the angular velocity to units of radians per second.

\begin{align*}
\omega & = \frac{5.00\ \text{rev}}{\text{sec}}\times \frac{2\pi \ \text{rad}}{1\ \text{rev}} \\ \\
\omega & = 31.4159 \ \text{rad/sec} \\ \\
\omega & = 31.4 \ \text{rad/sec}\ \qquad \ \color{DarkOrange} \left( \text{Answer} \right)
\end{align*}

Part B

For this part, we are asked to solve for the centripetal acceleration. We are going to use the formula a_{c} = r \omega ^2 given r=0.120\ \text{m} and \omega = 31.4159 \ \text{rad/s} .

\begin{align*}
a_{c} & = r \omega ^2 \\ \\ 
a_{c} & = \left( 0.120 \ \text{m} \right) \left( 31.4159 \ \text{rad/s} \right)^2 \\ \\
a_{c} & = 118.4350 \ \text{m/s}^2 \\ \\
a_{c} & = 118 \ \text{m/s}^2\ \qquad \ \color{DarkOrange} \left( \text{Answer} \right)
\end{align*}

Part C

For this part, we are going to directly solve the centripetal acceleration.

\begin{align*}
a_{c} & = r \omega ^2 \\ \\ 
a_{c} & = \left( 0.120 \ \text{m} \right)\left( \frac{9\ \text{rev}}{\text{s}} \times \frac{2\pi \ \text{rad}}{1\ \text{rev}}\right)^2 \\ \\
a_{c} & = 383.7302 \ \text{m/s}^2 \\ \\
a_{c} & = 384 \ \text{m/s}^2 \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right)
\end{align*}

Part D

The centripetal acceleration felt by Olympic skaters is 12 times larger than the acceleration due to gravity. That is quite a lot of acceleration in itself. The centripetal acceleration felt by Button’s nose was 39.2 times larger than the acceleration due to gravity! It is no wonder that he ruptured small blood vessels in his spins.


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Problem 6-15: The centripetal acceleration at the tip of a helicopter blade


Helicopter blades withstand tremendous stresses. In addition to supporting the weight of a helicopter, they are spun at rapid rates and experience large centripetal accelerations, especially at the tip.

(a) Calculate the magnitude of the centripetal acceleration at the tip of a 4.00 m long helicopter blade that rotates at 300 rev/min.

(b) Compare the linear speed of the tip with the speed of sound (taken to be 340 m/s).


Solution:

Part A

We are given the following values: r=4.00\ \text{m}, and \omega = 300 \ \text{rev/min}.

Let us convert the angular velocity to unit of radians per second.

\omega = 300 \  \frac{\text{rev}}{\text{min}} \times \frac{2\pi \ \text{rad}}{1 \ \text{rev}}\times \frac{1\ \text{min}}{60 \ \text{sec}} = 31.4159 \ \text{rad/sec}

The centripetal acceleration at the tip of the helicopter blade can be computed using the formula

a_{c} = r \omega ^2

If we substitute the given values into the formula, we have

\begin{align*}
a_{c} & = r \omega^2 \\ \\
a_{c} & = \left( 4.00\ \text{m} \right)\left( 31.4159 \ \text{rad/sec} \right)^2 \\ \\
a_{c} & = 3947.8351 \ \text{m/s}^2 \\ \\
a_{c} & = 3.95 \times10^3 \ \text{m/s}^2 \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right)
\end{align*}

Part B

We are asked to solve for the linear velocity of the blade’s tip. We are going to use the formula

v=r \omega

We just needed to substitute the given values into the formula.

\begin{align*}
v & = r \omega \\ \\
v & = \left( 4.00 \ \text{m} \right)\left( 31.4159 \ \text{rad/sec} \right) \\ \\
v & = 125.6636 \ \text{m/s} \\ \\
v & = 126 \ \text{m/s} \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right)
\end{align*}

Let us compare this with the speed of light which is 340 m/s.

\frac{125.6636 \ \text{m/s}}{340\ \text{m/s}} \times 100 \%= 36.9599 \% =37.0\%

The linear velocity of the blades tip is 37.0% of the speed of light.


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Problem 6-13: The motion of the WWII fighter plane propeller


The propeller of a World War II fighter plane is 2.30 m in diameter.

(a) What is its angular velocity in radians per second if it spins at 1200 rev/min?

(b) What is the linear speed of its tip at this angular velocity if the plane is stationary on the tarmac?

(c) What is the centripetal acceleration of the propeller tip under these conditions? Calculate it in meters per second squared and convert to multiples of g.


Solution:

Part A

We are converting the angular velocity \omega = 1200\ \text{rev/min} into radians per second.

\begin{align*}
\omega = & \frac{1200\ \text{rev}}{\text{min}}\times \frac{2\pi \ \text{radian}}{1\ \text{rev}} \times \frac{1 \ \text{min}}{60 \ \text{sec}} \\ \\
\omega = & 125.6637 \ \text{radians/sec} \\ \\
\omega = & 126 \ \text{radians/sec} \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right)
\end{align*}

Part B

We are now solving the linear speed of the tip of the propeller by relating the angular velocity to linear velocity using the formula v = r \omega . The radius is half the diameter, so r= \frac{2.30\ \text{m}}{2} = 1.15 \ \text{m} .

\begin{align*}
v & = r \omega \\ \\
v & = \left( 1.15 \ \text{m} \right)\left( 125.6637 \ \text{radians/sec} \right) \\ \\
v & = 144.5132 \ \text{m/s} \\ \\
v & = 145 \ \text{m/s} \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right)
\end{align*}

Part C

From the computed linear speed and the given radius of the propeller, we can now compute for the centripetal acceleration a_{c} using the formula

a_{c} = \frac{v^2}{r}

If we substitute the given values, we have

\begin{align*}
a_{c} & = \frac{v^2}{r} \\ \\
a_{c} & = \frac{\left( 144.5132 \ \text{m/s} \right)^2}{1.15 \ \text{m}} \\ \\
a_{c} & = 18160.0565 \ \text{m/s}^2 \\ \\
a_{c} & = 1.82\times 10^{4} \ \text{m/s}^2 \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right)
\end{align*}

We can convert this value in multiples of g

\begin{align*}
a_{c} & = 18160.0565 \ \text{m/s}^2 \times \frac{g}{9.81 \ \text{m/s}^2} \\ \\
a_{c} & = 1851.1780 g \\ \\
a_{c} & = 1.85\times 10^{3} \ g \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right)
\end{align*}

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Problem 6-12: The approximate total distance traveled by planet Earth since its birth


Taking the age of Earth to be about 4×109 years and assuming its orbital radius of 1.5 ×1011 m has not changed and is circular, calculate the approximate total distance Earth has traveled since its birth (in a frame of reference stationary with respect to the Sun).


Solution:

First, we need to compute for the linear velocity of the Earth using the formula below knowing that the Earth has 1 full revolution in 1 year

v=r\omega

where r=1.5\times 10^{11} \ \text{m} and \omega = 2\pi \ \text{rad/year} . Substituting these values, we have

\begin{align*}
v & = r \omega \\ \\
v & = \left( 1.5\times 10^{11} \ \text{m} \right)\left( 2 \pi \ \text{rad/year} \right) \\ \\
v & = 9.4248\times 10^{11} \ \text{m/year}
\end{align*}

Knowing the linear velocity, we can compute for the total distance using the formula

\Delta x = v \Delta t

We can now substitute the given values: v = 9.4248\times 10^{11} \ \text{m/year} and \Delta t = 4\times 10^{9} \ \text{years} .

\begin{align*}
\Delta x & = v \Delta t \\ \\
\Delta x & = \left( 9.4248\times 10^{11} \ \text{m/year}  \right) \left( 4\times 10^{9} \ \text{years} \right) \\ \\
\Delta x & = 3.7699 \times 10^{21} \ \text{m} \\ \\
\Delta x & = 4 \times 10^{21} \ \text{m} \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right)
\end{align*}

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Problem 6-11: Calculating the centripetal acceleration of a runner in a circular track


A runner taking part in the 200 m dash must run around the end of a track that has a circular arc with a radius of curvature of 30 m. If the runner completes the 200 m dash in 23.2 s and runs at constant speed throughout the race, what is the magnitude of their centripetal acceleration as they run the curved portion of the track?


Solution:

Centripetal acceleration a_{c} is the acceleration experienced while in uniform circular motion. It always points toward the center of rotation. It is perpendicular to the linear velocity v and has the magnitude

a_{c}=\frac{v^{2}}{r}

We can solve for the constant speed of the runner using the formula

v=\frac{\Delta x}{\Delta t}

We are given the distance \Delta x = 200 \ \text{m} , and the total time \Delta t = 23.2\ \text{s} . Therefore, the velocity is

\begin{align*}
v & =\frac{\Delta x}{\Delta t} \\ \\ 
v & = \frac{200\ \text{m}}{23.2\ \text{s}} \\ \\
v & = 8.6207\ \text{m/s}
\end{align*}

From the given problem, we are given the following values: r=30\ \text{m} . We now have the details to solve for the centripetal acceleration.

\begin{align*}
a_{c} & = \frac{v^{2}}{r} \\ \\
a_{c} & = \frac{\left( 8.6207\ \text{m/s} \right)^2}{30\ \text{m}} \\ \\
a_{c} & = 2.4772\ \text{m/s}^{2} \\ \\
a_{c} & = 2.5\  \text{m/s}^{2} \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right)
\end{align*}

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Problem 6-10: The angular velocity of a person in a circular fairground ride


A fairground ride spins its occupants inside a flying saucer-shaped container. If the horizontal circular path the riders follow has an 8.00 m radius, at how many revolutions per minute will the riders be subjected to a centripetal acceleration whose magnitude is 1.50 times that due to gravity?


Solution:

Centripetal acceleration a_{c} is the acceleration experienced while in uniform circular motion. It always points toward the center of rotation. The relationship between the centripetal acceleration a_{c} and the angular velocity \omega is given by the formula

a_{c}=r\omega^{2}

Now, taking the formula and solving for the angular velocity:

\omega = \sqrt{\frac{a_{c}}{r}}

From the given problem, we are given the following values: r=8.00\ \text{m} and a_{c}=1.50\times 9.81 \ \text{m/s}^2=14.715\ \text{m/s}^2. If we substitute these values in the formula, we can solve for the angular velocity.

\begin{align*}
\omega & = \sqrt{\frac{a_{c}}{r}} \\ \\
\omega & = \sqrt{\frac{14.715\ \text{m/s}^2}{8.00\ \text{m}}} \\ \\
\omega & = 1.3561\ \text{rad/sec} \\ \\
\end{align*}

Then, we can convert this value into its corresponding value at the unit of revolutions per minute.

\begin{align*}
\omega & = 1.3561\ \frac{\text{rad}}{\text{sec}} \times \frac{60\ \text{sec}}{1\ \text{min}}\times \frac{1\ \text{rev}}{2\pi \ \text{rad}} \\ \\
\omega & = 12.9498\ \text{rev/min} \\ \\
\omega & = 13.0 \ \text{rev/min} \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right)
\end{align*}

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Problem 6-8: An integrated problem involving circular motion, momentum, and projectile motion


Integrated Concepts

When kicking a football, the kicker rotates his leg about the hip joint.

(a) If the velocity of the tip of the kicker’s shoe is 35.0 m/s and the hip joint is 1.05 m from the tip of the shoe, what is the shoe tip’s angular velocity?

(b) The shoe is in contact with the initially stationary 0.500 kg football for 20.0 ms. What average force is exerted on the football to give it a velocity of 20.0 m/s?

(c) Find the maximum range of the football, neglecting air resistance.


Solution:

Part A

From the given problem, we are given the following values: v=35.0\ \text{m/s} and r=1.05\ \text{m}. We are required to solve for the angular velocity \omega.

The linear velocity, v and the angular velocity, \omega are related by the equation

v=r\omega \ \text{or} \ \omega=\frac{v}{r}

If we substitute the given values into the formula, we can directly solve for the value of the angular velocity. That is,

\begin{align*}
\omega & = \frac{v}{r} \\ \\
\omega & = \frac{35.0\ \text{m/s}}{1.05\ \text{m}} \\ \\
\omega & = 33.3333\ \text{rad/sec} \\ \\
\omega & = 33.3 \ \text{rad/s} \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right)
\end{align*}

Part B

For this part of the problem, we are going to use Newton’s second law of motion in term of linear momentum which states that the net external force equals the change in momentum of a system divided by the time over which it changes. That is

F_{net} = \frac{\Delta p}{\Delta t} = \frac{m\left( v_f - v_i \right)}{t}

For this problem, we are given the following values: m=0.500\ \text{kg}, t=20.0\times 10^{-3} \ \text{s}, v_{f}=20.0\ \text{m/s}, and v_{i}=0. Substituting all these values into the equation, we can solve directly for the value of the net external force.

\begin{align*}
F_{net} & = \frac{\left( 0.500\ \text{kg} \right)\left( 20.0\ \text{m/s}-0\ \text{m/s} \right)}{20.0\times 10^{-3}\ \text{s}} \\ \\
F_{net} & = 500\ \text{N} \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right)
\end{align*}

Part C

This is a problem on projectile motion. In this particular case, we are solving for the range of the projectile. The formula for the range of a projectile is

R=\frac{v_{0}^2 \sin 2\theta}{g}

We are asked to solve for the maximum range, and we know that the maximum range happens when the angle \theta is 45^\circ .

\begin{align*}
R & = \frac{\left( 20.0\ \text{m/s} \right)^{2} \sin \left( 2\left( 45^\circ  \right) \right)}{9.81 \ \text{m/s}^2} \\ \\
R & = 40.7747\ \text{m} \\ \\
R & = 40.8 \ \text{m} \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right)
\end{align*}

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