College Physics 2013

You have a pendulum pulled to the side, a heavy beach ball pushed partly under water, and a medicine ball lifted some height above the ground. Each object is then released. (a) Draw a picture of the motion of each object and a motion diagram for each. (b) Indicate the equilibrium position. (c) Draw two or more force diagrams at key points in the motion. (d) Use (a) through (c) to reason whether the motions of these objects can be considered vibrational motions.

Consider the three objects described in Problem 1. (a) Choose a system for each and construct energy bar charts for each motion. What initial and final states did you choose? (b) What assumptions did you make?

Exercise stretch cord You want to determine the spring constant of an exercise stretch cord. You pull the cord with a force probe that exerts a 50-N force on the cord, causing it to stretch 20 cm. (a) What is the spring constant of the cord? Describe your reasoning and assumptions. (b) How can you test your answer?

Two exercise cords In order to increase resistance, you put two exercise cords together as shown in Figure P19.4. What is the effective spring constant of the two bands compared to one? Use data from
Problem 3. Explain.

You have a ball bearing and a bowl. You let the ball roll down from the top of the bowl; it moves up and down the bowl’s wall for some time. (a) Is this vibrational motion? Explain why or why not. (b) Draw a force diagram for the ball for four different positions and indicate what force or force component is responsible for the acceleration toward the equilibrium position. (c) Draw energy bar charts for these four ball-Earth system positions.

(a) Draw a sketch of an object attached to a vertical spring. Indicate the equilibrium position. (b) Show this object vibrating up and down and indicate the amplitude of vibrations on the sketch. (c) Describe an experiment you would perform to determine the period of vibration of the object. (d) Perform the experiment. Include experimental uncertainty in your result.

Draw a sketch of a pendulum. Indicate the equilibrium position. Show this object vibrating and indicate the amplitude of vibration on the sketch.

Concert $A$ and $O_{2}$ vibration (a) Musicians in an orchestra tune their instruments to what is called "concert $A, "$ a frequency of 440 $\mathrm{Hz}$ . Determine the period for one vibration. (b) The atoms in an oxygen molecule complete one vibration in a time interval of $2.11 \times 10^{-14}$ s. Determine the frequency of vibration of $\mathrm{O}_{2}$ .

Bl0 Hearing range A doctor is checking your hearing. (a) If the period of the lowest-frequency sound you can hear is 0.050 $\mathrm{s}$ , then what is its frequency? (b) If the highest-frequency sound you can hear is $20,000 \mathrm{Hz}$ , then what is its period?

Draw a graph showing the position-versus-time curve for a simple harmonic oscillator (a) with twice the frequency of that shown in Figure P19.10 and (b) with the same frequency but twice the amplitude as shown in the figure.

Suppose that at time zero the cart attached to the spring such as shown in Figure 19.8 is released from rest at position $x=+A$ and that its period of vibration is 8.0 $\mathrm{s}$ . Draw nine sketches showing the cart's approximate position each second starting at 0 s and ending at 8.0 s. Draw and label arrows indicating the relative velocity and acceleration of the cart at each position.

(a) Sketch a motion diagram and a position-versus-time graph for the motion of a cart attached to a spring during one period. It passes at high speed through the equilibrium position at time zero. (b) Sketch a velocity-versus-time graph for the cart. (c) Sketch an acceleration-versus-time graph for the
cart. (d) Draw another representation of the process; explain how this representation allows you to learn more about the process than the kinematics graphs do.

The motion of a cart is described as $x=$ $(0.17 \mathrm{m}) \sin \left(\pi s^{-1}\right) t .$ Say everything you can about this motion and represent it using graphs, motion diagrams, and energy bar charts.

Sketch an acceleration-versus-time graph for the simple harmonic motion of an object of your choice. Indicate the amplitude of the acceleration and the period of the vibration. Underneath the graph draw corresponding position-versus- time and velocity-versus-time graphs with the correct shape but not necessarily the correct amplitudes.

Devise a position-versus-time function that describes the simple harmonic motion of an object of your choice. Choose the amplitude and the period of vibration, and draw corresponding velocity-versus-time and acceleration-versus time graphs. Draw a motion diagram for one period of its motion.

* The position of a vibrating object changes as a function of time as $x=(0.2 \mathrm{m}) \cos \left(\pi s^{-1}\right) t$ . Say everything you can about this motion. Write an expression for the velocity and acceleration as functions of time. Draw graphs for each function.

The position of a vibrating object changes as a function of of time as $v=-(0.6 \mathrm{m} / \mathrm{s}) \cos (2 \pi) .$ (a) Say everything you can about the motion. Draw a sketch of the situation when the motion starts. (b) Represent the motion with a position- versus-time graph and a work-energy bar chart.

$*$ A cart at the end of a spring undergoes simple harmonic motion of amplitude 10 $\mathrm{cm}$ and frequency 5.0 $\mathrm{Hz}$ . (a) Determine the period of vibration. (b) Write an expression for the cart's position at different times, assuming it is at $x=-A$ when $t=0 .$ Determine its position $(\mathrm{c})$ at 0.050 $\mathrm{s}$ and (d) at 0.100 $\mathrm{s}$ .

You exert a $100-\mathrm{N}$ pull on the end of a spring. When you in- crease the force by 20$\%$ to $120 \mathrm{N},$ the spring's length increases 5.0 $\mathrm{cm}$ beyond its original stretched position. What is the spring constant of the spring and its original displacement?

Metronome You want to make a metronome for music practice. You use a $30-\mathrm{g}$ object attached to a spring to serve as the time standard. What is the desired spring constant of the spring if the object needs to make 1.00 vibrations each second? What are the assumptions that you made?

Determine the frequency of vibration of the cart shown in Figure P19.21.

$*$ A spring with a cart at its end vibrates at frequency 6.0 $\mathrm{Hz}$ . (a) Determine the period of vibration. (b) Determine the frequency if the cart's mass is doubled while the spring constant remains unchanged and (c) the frequency if the spring constant doubles while the cart's mass remains the same.

A cart with mass $m$ vibrating at the end of a spring has an extra block added to it when its displacement is $x=+A .$ What should the block's mass be in order to reduce the frequency to half its initial value?

$*$ A $2.0-\mathrm{kg}$ cart vibrates at the end of an $18-\mathrm{N} / \mathrm{m}$ spring. (a) Make a list of physical quantities you can determine about the vibrations and determine two of them. (b) If a second $18-\mathrm{N} / \mathrm{m}$ spring is attached beside the first one, what will be the period
of the vibration?

$*$ What were the main ideas that we used to derive the expression for the period of an object vibrating at the end of a spring? Explain the assumptions that we made. When is the expression for the period not valid for a spring of known $k$ ?

$*$ A spring with a spring constant of 1200 $\mathrm{N} / \mathrm{m}$ has a $55-\mathrm{g}$ ball
at its end. (a) If the energy of the system is $6.0 \mathrm{J},$ what is the amplitude of vibration? (b) What is the maximum speed of the ball? (c) What is the speed when the ball is at a position $x=+A / 2 ?$ What assumptions did you make to solve the problem? How do you know if your answer makes sense?

A person exerts a $15-\mathrm{N}$ force on a cart attached to a spring and holds the cart steady. The cart is displaced 0.060 $\mathrm{m}$ from its equilibrium position. When the person stops holding
the cart, the system cart + spring undergoes simple harmonic motion. (a) Determine the spring constant of the spring. (b) Determine the energy of the system. (c) Write an expression $x(t)$ for the motion of the cart. (d) Draw as many graphical representations of the motion as you can.

$*$ A spring with spring constant $2.5 \times 10^{4} \mathrm{N} / \mathrm{m}$ has a $1.4-\mathrm{kg}$ cart at its end. (a) If its amplitude of vibration is $0.030 \mathrm{m},$ what is the total energy of the cart $+$ spring system? (b) What is the maximum speed of the cart? (c) If the energy is tripled, what is the new amplitude? (d) What is the maximum speed of the cart? (e) What assumptions did you make to solve the problem? If the assumptions were not reasonable, how would the answers change?

$*$ Proportional reasoning By what factor must we increase the amplitude of vibration of an object at the end of a spring in order to double its maximum speed during a vibration? Explain.

Proportional reasoning By what factor must we increase the amplitude of vibration of an object at the end of a spring in order to double the total energy of the system? Explain. What will happen to the speed the object? What assumptions did you make?

Monkey trick at zoo A monkey has a cart with a horizontal spring attached to it that she uses for different tricks. In one trick, the monkey sits on the vibrating cart. When the cart reaches its maximum displacement from equilibrium, the monkey picks up a 0.30-kg cantaloupe from a trainer. The mass of the monkey and the cart together is 3.0 kg. The spring constant is 660 N/m. The amplitude of horizontal vibrations is 0.24 m. Determine the ratio of the maximum speed of the monkey before and after she picks up the cantaloupe.

A cart attached to a spring vibrates with amplitude $A$ (a) What fraction of the total energy of the cart spring system is elastic potential energy and what fraction is kinetic energy when the cart is at position $x=A / 2 ?$ (b) At what position is the cart when its kinetic energy equals its elastic potential energy?

A 2.0 -kg cart attached to a spring undergoes simple harmonic motion so that its displacement is described by $x=(0.20 \mathrm{m}) \sin [(2 \pi / 2.0 \mathrm{s}) t] .$ Construct energy bar charts for the cart-spring system at times $t=0, t=T / 4$ $t=T / 2, t=3 T / 4,$ and $t=T$

$* /$ Equation Jeopardy The following expression describes a situation involving vibrational motion. Sketch a process and devise a problem for which the expression might be an answer.
$$
\begin{aligned} \frac{1}{2}(20,000 \mathrm{N} / \mathrm{m})(0.20 \mathrm{m})^{2}=& \frac{1}{2}(100 \mathrm{kg}) v^{2} \\ &+\frac{1}{2}(20,000 \mathrm{N} / \mathrm{m})(0.10 \mathrm{m})^{2} \end{aligned}
$$

Pendulum clock Shawn wants to build a clock whose pendulum makes one swing back and forth each second. (a) What is the desired length of the rod (assumed to have negligible mass) holding the metal ball at its end? (b) Will the rod need to be shorter or longer if he includes the mass of the rod in the
calculations? Explain.

Show that the expression for the frequency of a pendulum as a function of its length is dimensionally correct.

A pendulum swings with amplitude 0.020 m and period of 2.0 s. What is its maximum speed?

$*$ Proportional reasoning You are designing a pendulum clock whose period can be adjusted by 10$\%$ by changing the length of the pendulum. By what percent must you be able to change the length to provide this flexibility in the period? Explain.

Building demolition $A 500$ -kg ball at the end of a $30-m$ cable suspended from a crane is used to demolish an old building. If the ball has an initial angular displacement of $15^{\circ}$ from the vertical, determine its speed at the bottom of the arc.

$*$ You have a pendulum with a long string whose length you can vary but cannot measure, a small ball, and a stopwatch. Describe two experiments that you can design to determine the height of a ladder. Indicate the assumptions that you use in each method.

" Variations in $g$ The frequency of a person's pendulum is 0.3204 $\mathrm{Hz}$ when at a location where $g$ is known to be exactly 9.800 $\mathrm{m} / \mathrm{s}^{2} .$ Where might the same pendulum be when its frequency is 0.3196 $\mathrm{Hz}$ ? What is $g$ at that location?

A graph of position versus time for an object undergoing simple harmonic motion is shown in Figure
P19.42. Estimate from the graph the amplitude and period of the motion and determine the object’s frequency. If the object is a pendulum, what is its length?

Determine the period of a 1.3-m-long pendulum on the Moon.

$*$ Trampoline vibration When a $60-\mathrm{kg}$ boy sits at rest on a trampoline, it sags 0.10 $\mathrm{m}$ at the center. (a) What is the effective spring constant for the trampoline? (b) The trampoline is pulled downward an extra 0.050 $\mathrm{m}$ by a strap sewed under the center of the trampoline. When the strap is released, what are the energy and frequency of the trampoline? What assumptions did you make?

$*$ A 1.2 -kg block sliding at 6.0 $\mathrm{m} / \mathrm{s}$ on a frictionless surface runs into and sticks to a spring. The spring is compressed 0.10 $\mathrm{m}$ before stopping the block and starting its motion back in the opposite direction. What can you determine about the vibrations that start after the collision? Make a list of physical quantities and determine four of them.

$*$ Proportional reasoning If you double the amplitude of vi- bration of an object at the end of a spring, how does this affect the values of $k, T, U,$ and $v_{\max } ?$

$*$ EST Willis Tower vibration The mass of the Willis (formerly Sears) Tower in Chicago is about $2 \times 10^{8} \mathrm{kg}$ . The tower sways back and forth at a frequency of about 0.10 $\mathrm{Hz}$ . (a) Estimate the effective spring constant for this swaying motion. Explain why you included a particular number of significant figures. (b) A gust of wind hitting the building exerts a
force of about $4 \times 10^{6} \mathrm{N}$ . By approximately how much is the top of the building displaced by the wind? Explain why you included a particular number of significant figures. State the
assumptions that you made. Does your make sense?

* EST BIO Annoying sound Low-frequency vibrations (less than 5 $\mathrm{Hz}$ ) are annoying to humans if the product of the amplitude and the frequency squared $\left(A f^{2}\right)$ equals
$0.5 \times 10^{-2} \mathrm{m} \cdot \mathrm{s}^{-2}(\text { or more). Estimate the frequency and }$
amplitude of a buzzer that produces these annoying vibrations if the device vibrates a total mass of 0.12 $\mathrm{kg}$ and has a vibrational energy of 0.012 $\mathrm{J}$ .

$* *$ You shoot a 0.050 -kg arrow into a 0.50 -kg wooden cart that sits on a horizontal, frictionless surface at the end of a spring that is attached to the wall at the other end. The arrow hits the cart and sticks into it. The cart and arrow together compress the spring and start the system vibrating at a frequency of 2.0 Hz with a $0.20-$ -m amplitude. How fast was the arrow moving? State any assumptions you made to solve the problem.

$*$ Pendulum on Mars The frequency of a pendulum is 39$\%$ less when on Mars than when on Earth's surface. Use this fact to determine Mars's gravitational constant.

* This chapter stated that when damping is present, the period of vibration of a system is more than without damping. How can you test this assertion? Provide the details.

$*$ You have a pendulum whose length is 1.3 $\mathrm{m}$ and bob mass is 0.20 $\mathrm{kg}$ . The amplitude of vibration of the pendulum is 0.07 $\mathrm{m}$ . (a) Determine the maximum energy of the bob-Earth system. Explain why you included a particular number of significant figures. (b) How much mechanical energy is converted to internal energy before the pendulum stops?

You have a $0.10-\mathrm{kg}$ cart on a spring. The spring constant of the spring is 20 $\mathrm{N} / \mathrm{m}$ . The cart's initial vibration amplitude is 0.10 $\mathrm{m}$ . ( a Make a list of physical quantities you can determine using this information and determine three of them. (b) For approximately how long will the vibrations last if 10$\%$ of the mechanical energy during each cycle is converted into
internal energy during each cycle?

You have a spring that stretches 0.070 $\mathrm{m}$ when a 0.10 -kg block is attached to and hangs from it. Imagine that you slowly pull down with a spring scale so the block is now below the
equilibrium position where it was hanging at rest. The scale reading when you let go of the block is 3.0 $\mathrm{N}$ . (a) Where was the block when you let go? (b) Determine the work you did stretching the spring. (c) What was the energy of the spring- Earth system when you let go? (d) How far will the block rise after you release it? (e) The vibrations last for 50 cycles. Qualitatively represent the beginning of cycle $\# 1$ and beginning of cycle $\# 25$ with an energy bar chart.

Imagine that you have a cart on a spring that moves on a rough surface. (a) Represent the cart’s motion with a motion diagram for one period. (b) Draw force diagrams for each quarter of a period. (c) Draw energy bar charts for each quarter of a period. (d) Draw position-, velocity-, and acceleration-versus-time graphs for each period. (e) On these graphs, use a dashed line to sketch graphs for the same cart
on the same spring, assuming no friction between the cart and the surface.

Twins on a swing How frequently do you need to push a swing with twin brothers on it compared to when pushing the swing with one on it? What assumptions did you make?

$*$ (a) Determine the maximum speed of a girl on a $3.0-\mathrm{m}$ -long swing when the amplitude of vibration is 1.2 $\mathrm{m} .$ (b) What assumptions did you make? Is the child's vibrational motion
damped? Explain. (c) Under what conditions is the motion a forced vibration? Explain.

You have a 0.20 -kg block on a $10-\mathrm{N} / \mathrm{m}$ spring that oscillates up and down. How often do you need to push the block upward when it passes the bottom of its motion to increase the
amplitude of its vibrations?

Sloshing water You carry a bucket with water that has a natural sloshing period of 1.7 s. At what walking speed will water splash if your step is 0.90 $\mathrm{m}$ long?

$*$ Feeling road vibrations in a car If the average distance between bumps on a road is about 10 $\mathrm{m}$ and the natural frequency of the suspension system in the car is about 0.90 $\mathrm{Hz}$ , at what speed will you feel the bumps the most?

$*$ EST H atom vibration A hydrogen atom of mass $1.67 \times 10^{-27} \mathrm{kg}$ is attached to a very large protein by a bond that behaves much like a spring. (a) If the vibrational frequency of the hydrogen is $1.0 \times 10^{14} \mathrm{Hz}$ , what is the "effective" spring constant of this spring-like bond? (b) If the total vibrational energy is $k T(k \text { is Boltzmann's constant and } T \text { is }$ the temperature in kelvins), approximately what is the classical amplitude of vibration at room temperature $(T=300 \mathrm{K})$ ? By comparison, the diameter of a hydrogen atom is about $10^{-10} \mathrm{m} .$ State any assumptions that you made.

Child's bouncy chair You are designing a bouncy chair for a neighbor's child. A 1.0 -kg chair hangs from a spring that is attached to the ceiling. When you put the $10.0-\mathrm{kg}$ child in the chair, the system vibrates with a period of 3.0 $\mathrm{s}$ . What would be the period if the child's mother (mass 50 $\mathrm{kg} )$ sits on the same chair? What assumptions did you make?

$* /$ You attach a block (mass $m )$ to a spring (spring constant $k )$ that oscillates in a vertical direction. Determine an expression for the period of its vibrations. Sketch a position-versus-time graph for the vibrations, assuming that the motion starts when the block is at its lowest maximum displacement from the equilibrium position. What assumptions did youmake?

$*$ You attach a $1.6-\mathrm{kg}$ object to a spring, pull it down 0.12 $\mathrm{m}$ from the equilibrium position, and let it vibrate. You find that it takes 5.0 $\mathrm{s}$ for the object to complete 10 vibrations. Make a list of physical quantities that you can determine about the motion of the object and determine five of them.

$*$ Traveling through Earth A hole is drilled through the center of Earth. The gravitational force exerted by Earth on an object of mass $m$ as it goes through the hole is $m g(r / R)$ , where $r$ is the distance of the object from Earth's center and $R$ is the radius of Earth $\left(6.4 \times 10^{6} \mathrm{m}\right) .$ Will an object dropped into the hole execute simple harmonic motion? If yes, find the period of the motion. How does one-half this time compare with the time needed to fly in an airplane halfway around Earth? What assumptions did you make to solve the problem?

EST Estimate the effective spring constant of the suspension system of a car. Describe your technique carefully. How can you test your answer to determine if it makes sense?

* Use dimensional analysis to show that the expressions for the periods of an object attached to a spring and for a simple pendulum are reasonable.

* BIO Vibration amplitude in ear The weakest sound a human ear can possibly hear makes the ear vibrate with the energy of about $10^{-19} \mathrm{J}$ . If the spring constant of the ear is $20 \mathrm{N} / \mathrm{m},$ what is the amplitude of vibration of the eardrum? What assumptions did you make? How does this compare to the size of an atom (about $0.5 \times 10^{-10} \mathrm{m} ) ?$

$* *$ A 5.0 -g bullet traveling horizontally at an unknown speed hits and embeds itself in a $0.195-\mathrm{kg}$ block resting on a frictionless table. The block slides into and compresses a $180-\mathrm{N} / \mathrm{m}$ spring a distance of 0.10 $\mathrm{m}$ before stopping the block and bullet. Determine the initial speed of the bullet. State any assumptions that you made to solve the problem.

$* *$ You have a pendulum of mass $m$ and length $L$ that is displaced an angle $\theta$ at the start of the swinging. (a) Determine an expression for the energy of the bob-Earth system. (b) Determine an expression for the maximum vertical height of the bob with respect to the equilibrium position.
(c) Determine an expression for the maximum speed of the bob. (d) What assumptions did you make? (e) Discuss how the answer to each of the questions changes if relevant assumptions are not valid.

A pendulum clock works for many days before the swinging stops. Describe several mechanisms that allow an actual pendulum clock to continue working without decreasing amplitude. Write a report explaining how it works.

$* *$ Foucault's pendulum In 1851 , the French physicist Jean Foucault hung a large iron ball on a wire about 67 $\mathrm{m}(220 \mathrm{ft})$ long to show that Earth rotates. The pendulum appears to
continuously change the plane in which it swings as time elapses. Determine the swinging frequency of this pendulum. Explain why the behavior of Foucault's pendulum provides supporting evidence for the hypothesis pendulum provides ertial reference frame and helps reject the model of a geocentric universe (Earth at the center).

You push down on a raft floating on a lake. The raft sinks and then vibrates up and down for a short time. How does the period of its vibrations depend on the size and mass of the raft?

Which expression below represents the mass $m$ of Earth inside a sphere of radius $r$ smaller than the radius $R$ of Earth? Note that $\rho$ is the density of Earth, assumed uniform.
$$
\begin{array}{ll}{\text { (a) }(4 / 3) \pi r^{3} \rho} & {\text { (b) } G(4 / 3) \pi r^{3} \rho} \\ {\text { (c) } G M_{\text { Earth }} / r^{2}} & {\text { (d) } G(4 / 3) \pi r \rho} \\ {\text { (e) }m g} \end{array}
$$

Which expression below represents the magnitude of the re- storing force that Earth exerts on an object of mass $m$ when a distance $r$ from the center of Earth? Note that $\rho$ is the density of Earth, assumed uniform.
$$
\begin{array}{ll}{\text { (a) } m(4 / 3) \pi r^{3} \rho} & {\text { (b) } m G(4 / 3) \pi r^{3} \rho} \\ {\text { (c) } G M_{\mathrm{E}} m / r^{2}} & {\text { (d) } m G(4 / 3) \pi r \rho} \\ {\text { (e) } m g r}\end{array}
$$

Which expression below represents the period $T$ for one oscillation from Earth's surface through the center, to the other side, and then back again?
$\begin{array}{ll}{\text { (a) } 2 \pi(3 / G 4 \pi \rho)^{1 / 2}} & {\text { (b) } 2 \pi(G 4 \pi \rho / 3)^{1 / 2}} \\ {\text { (c) } 2 \pi(3 m / G 4 \pi \rho)^{1 / 2}} & {\text { (d) } 2 \pi(m G 4 \pi \rho / 3)^{1 / 2}} \\ {\text { (e) } 2 \pi(m / k)^{1 / 2}} \end{array}$

During the trip, when will your acceleration be greatest?
(a) At the beginning and end of the trip
(b) At the end of the trip
(c) When passing through the center of Earth
(d) The same for entire the trip

During the trip, when will your speed be greatest?
(a) At the beginning and ends of the trip
(b) At the end of the trip
(c) When passing through the center of Earth
(d) The same for entire the trip

If the radius of Earth is $6.4 \times 10^{\circ} \mathrm{m},$ its mass is $6.0 \times 10^{24} \mathrm{kg},$ and $G=6.67 \times 10^{-11} \mathrm{N} \cdot \mathrm{m}^{2} / \mathrm{kg}^{2},$ which answer is closest to the time interval for one trip through Earth?
$$\begin{array}{llll}{\text { (a) } 0.7 \mathrm{h}} & {\text { (b) } 1.4 \mathrm{h}} & {\text { (c) } 3.2 \mathrm{h}}\\{\text { (d) } 4.8 $\mathrm{h}}&{\text { (e) } 6.3 $\mathrm{h}}\end{array}$$

What causes vibrations in the fluid in the cochlea?
(a) The eardrum pushes against the cochlea.
(b) Sound waves push against the cochlea.
(c) A bone in the middle ear pushes against the cochlea.
(d) The fluid vibrates when something outside the ear vibrates.
(e) The fluid does not vibrate—the hair cells vibrate.

How does the basilar membrane distinguish different frequency vibrations?
(a) The hair cells have different lengths.
(b) The dimensions and stiffness of the cochlea vary along its length.
(c) The basilar membrane vibrates where the cochlea dimension and stiffness match the vibration frequency.
(d) The cochlea fluid resonates in only one part of the cochlea.
(e) All the hair cells bend back and forth at only the frequency of the vibration.

If you shake the board shown in Figure 19.20 at a frequency higher than the natural frequency of the rod on the right, then what happens?
(a) None of the rods vibrate.
(b) All of the rods vibrate.
(c) The shortest rod vibrates.
(d) The longest rod vibrates.
(e) The middle rods vibrate.

If you were to shake the special board (the one that has 15,000 rods of varying length) at one particular frequency, then what would happen?
(a) None of the rods would vibrate.
(b) All of the rods would vibrate.
(c) A small number of rods at one location would vibrate.
(d) A disturbance would travel back and forth along the board.

You hang four pendulum bobs from strings connected to a wooden dowel. The strings are different lengths. How can you get the second longest pendulum bob to vibrate while the other three do not—without touching the pendulums?
(a) Shake the dowel back and forth.
(b) Shake the dowel back and forth at the resonant frequency of that pendulum.
(c) Move the dowel sideways at any frequency.
(d) Blow air on that bob.