15 Oscillations
15 Chapter Review
Key Terms
- amplitude (A)
- maximum displacement from the equilibrium position of an object oscillating around the equilibrium position
- critically damped
- condition in which the damping of an oscillator causes it to return as quickly as possible to its equilibrium position without oscillating back and forth about this position
- elastic potential energy
- potential energy stored as a result of deformation of an elastic object, such as the stretching of a spring
- equilibrium position
- position where the spring is neither stretched nor compressed
- force constant (k)
- characteristic of a spring which is defined as the ratio of the force applied to the spring to the displacement caused by the force
- frequency (f)
- number of events per unit of time
- natural angular frequency
- angular frequency of a system oscillating in SHM
- oscillation
- single fluctuation of a quantity, or repeated and regular fluctuations of a quantity, between two extreme values around an equilibrium or average value
- overdamped
- condition in which damping of an oscillator causes it to return to equilibrium without oscillating; oscillator moves more slowly toward equilibrium than in the critically damped system
- period (T)
- time taken to complete one oscillation
- periodic motion
- motion that repeats itself at regular time intervals
- phase shift
- angle, in radians, that is used in a cosine or sine function to shift the function left or right, used to match up the function with the initial conditions of data
- physical pendulum
- any extended object that swings like a pendulum
- resonance
- large amplitude oscillations in a system produced by a small amplitude driving force, which has a frequency equal to the natural frequency
- restoring force
- force acting in opposition to the force caused by a deformation
- simple harmonic motion (SHM)
- oscillatory motion in a system where the restoring force is proportional to the displacement, which acts in the direction opposite to the displacement
- simple harmonic oscillator
- a device that oscillates in SHM where the restoring force is proportional to the displacement and acts in the direction opposite to the displacement
- simple pendulum
- point mass, called a pendulum bob, attached to a near massless string
- stable equilibrium point
- point where the net force on a system is zero, but a small displacement of the mass will cause a restoring force that points toward the equilibrium point
- torsional pendulum
- any suspended object that oscillates by twisting its suspension
- underdamped
- condition in which damping of an oscillator causes the amplitude of oscillations of a damped harmonic oscillator to decrease over time, eventually approaching zero
Key Equations
| Relationship between frequency and period | [latex]f=\frac{1}{T}[/latex] |
| Position in SHM with [latex]\varphi =0.00[/latex] | [latex]x(t)=A\,\text{cos}(\omega t)[/latex] |
| General position in SHM | [latex]x(t)=A\text{cos}(\omega t+\varphi )[/latex] |
| General velocity in SHM | [latex]v(t)=\text{−}A\omega \text{sin}(\omega t+\varphi )[/latex] |
| General acceleration in SHM | [latex]a(t)=\text{−}A{\omega }^{2}\text{cos}(\omega t+\varphi )[/latex] |
| Maximum displacement (amplitude) of SHM | [latex]{x}_{\text{max}}=A[/latex] |
| Maximum velocity of SHM | [latex]|{v}_{\text{max}}|=A\omega[/latex] |
| Maximum acceleration of SHM | [latex]|{a}_{\text{max}}|=A{\omega }^{2}[/latex] |
| Angular frequency of a mass-spring system in SHM | [latex]\omega =\sqrt{\frac{k}{m}}[/latex] |
| Period of a mass-spring system in SHM | [latex]T=2\pi \sqrt{\frac{m}{k}}[/latex] |
| Frequency of a mass-spring system in SHM | [latex]f=\frac{1}{2\pi }\sqrt{\frac{k}{m}}[/latex] |
| Energy in a mass-spring system in SHM | [latex]{E}_{\text{Total}}=\frac{1}{2}k{x}^{2}+\frac{1}{2}m{v}^{2}=\frac{1}{2}k{A}^{2}[/latex] |
| The velocity of the mass in a spring-mass system in SHM |
[latex]v=\pm\sqrt{\frac{k}{m}({A}^{2}-{x}^{2})}[/latex] |
| The x-component of the radius of a rotating disk | [latex]x(t)=A\text{cos}(\omega \,t+\varphi )[/latex] |
| The x-component of the velocity of the edge of a rotating disk | [latex]v(t)=\text{−}{v}_{\text{max}}\text{sin}(\omega \,t+\varphi )[/latex] |
| The x-component of the acceleration of the edge of a rotating disk |
[latex]a(t)=\text{−}{a}_{\text{max}}\text{cos}(\omega \,t+\varphi )[/latex] |
| Force equation for a simple pendulum | [latex]\frac{{d}^{2}\theta }{d{t}^{2}}=-\frac{g}{L}\theta[/latex] |
| Angular frequency for a simple pendulum | [latex]\omega =\sqrt{\frac{g}{L}}[/latex] |
| Period of a simple pendulum | [latex]T=2\pi \sqrt{\frac{L}{g}}[/latex] |
| Angular frequency of a physical pendulum | [latex]\omega =\sqrt{\frac{mgL}{I}}[/latex] |
| Period of a physical pendulum | [latex]T=2\pi \sqrt{\frac{I}{mgL}}[/latex] |
| Period of a torsional pendulum | [latex]T=2\pi \sqrt{\frac{I}{\kappa }}[/latex] |
| Newton’s second law for harmonic motion | [latex]m\frac{{d}^{2}x}{d{t}^{2}}+b\frac{dx}{dt}+kx=0[/latex] |
| Solution for underdamped harmonic motion | [latex]x(t)={A}_{0}{e}^{-\frac{b}{2m}t}\text{cos}(\omega t+\varphi )[/latex] |
| Natural angular frequency of a mass-spring system |
[latex]{\omega }_{0}=\sqrt{\frac{k}{m}}[/latex] |
| Angular frequency of underdamped harmonic motion |
[latex]\omega =\sqrt{{\omega }_{0}^{2}-{(\frac{b}{2m})}^{2}}[/latex] |
| Newton’s second law for forced, damped oscillation |
[latex]\text{−}kx-b\frac{dx}{dt}+{F}_{o}\text{sin}(\omega t)=m\frac{{d}^{2}x}{d{t}^{2}}[/latex] |
| Solution to Newton’s second law for forced, damped oscillations |
[latex]x(t)=A\text{cos}(\omega t+\varphi )[/latex] |
| Amplitude of system undergoing forced, damped oscillations |
[latex]A=\frac{{F}_{o}}{\sqrt{m{({\omega }^{2}-{\omega }_{o}^{2})}^{2}+{b}^{2}{\omega }^{2}}}[/latex] |
Summary
15.1 Simple Harmonic Motion
- Periodic motion is a repeating oscillation. The time for one oscillation is the period T and the number of oscillations per unit time is the frequency f. These quantities are related by f=1Tf=1T.
- Simple harmonic motion (SHM) is oscillatory motion for a system where the restoring force is proportional to the displacement and acts in the direction opposite to the displacement.
- Maximum displacement is the amplitude A. The angular frequency ωω, period T, and frequency f of a simple harmonic oscillator are given by ω=km‾‾√ω=km, T=2πmk‾‾√,andf=12πkm‾‾√T=2πmk,andf=12πkm, where m is the mass of the system and k is the force constant.
- Displacement as a function of time in SHM is given byx(t)=Acos(2πTt+ϕ)=Acos(ωt+ϕ)x(t)=Acos(2πTt+ϕ)=Acos(ωt+ϕ).
- The velocity is given by v(t)=−Aωsin(ωt+ϕ)=−vmaxsin(ωt+ϕ),wherevmax=Aω=Akm‾‾√v(t)=−Aωsin(ωt+ϕ)=−vmaxsin(ωt+ϕ),wherevmax=Aω=Akm.
- The acceleration is a(t)=−Aω2cos(ωt+ϕ)=−amaxcos(ωt+ϕ)a(t)=−Aω2cos(ωt+ϕ)=−amaxcos(ωt+ϕ), where amax=Aω2=Akmamax=Aω2=Akm.
15.2 Energy in Simple Harmonic Motion
- The simplest type of oscillations are related to systems that can be described by Hooke’s law, F = −kx, where F is the restoring force, x is the displacement from equilibrium or deformation, and k is the force constant of the system.
- Elastic potential energy U stored in the deformation of a system that can be described by Hooke’s law is given byU=12kx2.U=12kx2.
- Energy in the simple harmonic oscillator is shared between elastic potential energy and kinetic energy, with the total being constant:
ETotal=12mv2+12kx2=12kA2=constant.ETotal=12mv2+12kx2=12kA2=constant.
- The magnitude of the velocity as a function of position for the simple harmonic oscillator can be found by using
∣∣v∣∣=km(A2−x2)‾‾‾‾‾‾‾‾‾‾‾‾√.|v|=km(A2−x2).
15.3 Comparing Simple Harmonic Motion and Circular Motion
- A projection of uniform circular motion undergoes simple harmonic oscillation.
- Consider a circle with a radius A, moving at a constant angular speed ωω. A point on the edge of the circle moves at a constant tangential speed of vmax=Aωvmax=Aω. The projection of the radius onto the x-axis is x(t)=Acos(ωt+ϕ)x(t)=Acos(ωt+ϕ), where (ϕ)(ϕ) is the phase shift. The x-component of the tangential velocity is v(t)=−Aωsin(ωt+ϕ)v(t)=−Aωsin(ωt+ϕ).
15.4 Pendulums
- A mass m suspended by a wire of length L and negligible mass is a simple pendulum and undergoes SHM for amplitudes less than about 15°15°. The period of a simple pendulum is T=2πLg‾‾√T=2πLg, where L is the length of the string and g is the acceleration due to gravity.
- The period of a physical pendulum T=2πImgL‾‾‾‾√T=2πImgL can be found if the moment of inertia is known. The length between the point of rotation and the center of mass is L.
- The period of a torsional pendulum T=2πIκ‾‾√T=2πIκ can be found if the moment of inertia and torsion constant are known.
15.5 Damped Oscillations
- Damped harmonic oscillators have non-conservative forces that dissipate their energy.
- Critical damping returns the system to equilibrium as fast as possible without overshooting.
- An underdamped system will oscillate through the equilibrium position.
- An overdamped system moves more slowly toward equilibrium than one that is critically damped.
15.6 Forced Oscillations
- A system’s natural frequency is the frequency at which the system oscillates if not affected by driving or damping forces.
- A periodic force driving a harmonic oscillator at its natural frequency produces resonance. The system is said to resonate.
- The less damping a system has, the higher the amplitude of the forced oscillations near resonance. The more damping a system has, the broader response it has to varying driving frequencies.
Conceptual Questions
15.1 Simple Harmonic Motion
1.
What conditions must be met to produce SHM?
2.
(a) If frequency is not constant for some oscillation, can the oscillation be SHM? (b) Can you think of any examples of harmonic motion where the frequency may depend on the amplitude?
3.
Give an example of a simple harmonic oscillator, specifically noting how its frequency is independent of amplitude.
4.
Explain why you expect an object made of a stiff material to vibrate at a higher frequency than a similar object made of a more pliable material.
5.
As you pass a freight truck with a trailer on a highway, you notice that its trailer is bouncing up and down slowly. Is it more likely that the trailer is heavily loaded or nearly empty? Explain your answer.
6.
Some people modify cars to be much closer to the ground than when manufactured. Should they install stiffer springs? Explain your answer.
15.2 Energy in Simple Harmonic Motion
7.
Describe a system in which elastic potential energy is stored.
8.
Explain in terms of energy how dissipative forces such as friction reduce the amplitude of a harmonic oscillator. Also explain how a driving mechanism can compensate. (A pendulum clock is such a system.)
9.
The temperature of the atmosphere oscillates from a maximum near noontime and a minimum near sunrise. Would you consider the atmosphere to be in stable or unstable equilibrium?
15.3 Comparing Simple Harmonic Motion and Circular Motion
10.
Can this analogy of SHM to circular motion be carried out with an object oscillating on a spring vertically hung from the ceiling? Why or why not? If given the choice, would you prefer to use a sine function or a cosine function to model the motion?
11.
If the maximum speed of the mass attached to a spring, oscillating on a frictionless table, was increased, what characteristics of the rotating disk would need to be changed?
15.4 Pendulums
12.
Pendulum clocks are made to run at the correct rate by adjusting the pendulum’s length. Suppose you move from one city to another where the acceleration due to gravity is slightly greater, taking your pendulum clock with you, will you have to lengthen or shorten the pendulum to keep the correct time, other factors remaining constant? Explain your answer.
13.
A pendulum clock works by measuring the period of a pendulum. In the springtime the clock runs with perfect time, but in the summer and winter the length of the pendulum changes. When most materials are heated, they expand. Does the clock run too fast or too slow in the summer? What about the winter?
14.
With the use of a phase shift, the position of an object may be modeled as a cosine or sine function. If given the option, which function would you choose? Assuming that the phase shift is zero, what are the initial conditions of function; that is, the initial position, velocity, and acceleration, when using a sine function? How about when a cosine function is used?
15.5 Damped Oscillations
15.
Give an example of a damped harmonic oscillator. (They are more common than undamped or simple harmonic oscillators.)
16.
How would a car bounce after a bump under each of these conditions?
(a) overdamping
(b) underdamping
(c) critical damping
17.
Most harmonic oscillators are damped and, if undriven, eventually come to a stop. Why?
15.6 Forced Oscillations
18.
Why are soldiers in general ordered to “route step” (walk out of step) across a bridge?
19.
Do you think there is any harmonic motion in the physical world that is not damped harmonic motion? Try to make a list of five examples of undamped harmonic motion and damped harmonic motion. Which list was easier to make?
20.
Some engineers use sound to diagnose performance problems with car engines. Occasionally, a part of the engine is designed that resonates at the frequency of the engine. The unwanted oscillations can cause noise that irritates the driver or could lead to the part failing prematurely. In one case, a part was located that had a length L made of a material with a massM. What can be done to correct this problem?
Problems
15.1 Simple Harmonic Motion
21.
Prove that using x(t)=Asin(ωt+ϕ)x(t)=Asin(ωt+ϕ) will produce the same results for the period for the oscillations of a mass and a spring. Why do you think the cosine function was chosen?
22.
What is the period of 60.0 Hz of electrical power?
23.
If your heart rate is 150 beats per minute during strenuous exercise, what is the time per beat in units of seconds?
24.
Find the frequency of a tuning fork that takes 2.50×10−3s2.50×10−3s to complete one oscillation.
25.
A stroboscope is set to flash every 8.00×10−5s8.00×10−5s. What is the frequency of the flashes?
26.
A tire has a tread pattern with a crevice every 2.00 cm. Each crevice makes a single vibration as the tire moves. What is the frequency of these vibrations if the car moves at 30.0 m/s?
27.
Each piston of an engine makes a sharp sound every other revolution of the engine. (a) How fast is a race car going if its eight-cylinder engine emits a sound of frequency 750 Hz, given that the engine makes 2000 revolutions per kilometer? (b) At how many revolutions per minute is the engine rotating?
28.
A type of cuckoo clock keeps time by having a mass bouncing on a spring, usually something cute like a cherub in a chair. What force constant is needed to produce a period of 0.500 s for a 0.0150-kg mass?
29.
A mass m0m0 is attached to a spring and hung vertically. The mass is raised a short distance in the vertical direction and released. The mass oscillates with a frequency f0f0. If the mass is replaced with a mass nine times as large, and the experiment was repeated, what would be the frequency of the oscillations in terms of f0f0 ?
30.
A 0.500-kg mass suspended from a spring oscillates with a period of 1.50 s. How much mass must be added to the object to change the period to 2.00 s?
31.
By how much leeway (both percentage and mass) would you have in the selection of the mass of the object in the previous problem if you did not wish the new period to be greater than 2.01 s or less than 1.99 s?
15.2 Energy in Simple Harmonic Motion
32.
Fish are hung on a spring scale to determine their mass. (a) What is the force constant of the spring in such a scale if it the spring stretches 8.00 cm for a 10.0 kg load? (b) What is the mass of a fish that stretches the spring 5.50 cm? (c) How far apart are the half-kilogram marks on the scale?
33.
It is weigh-in time for the local under-85-kg rugby team. The bathroom scale used to assess eligibility can be described by Hooke’s law and is depressed 0.75 cm by its maximum load of 120 kg. (a) What is the spring’s effective force constant? (b) A player stands on the scales and depresses it by 0.48 cm. Is he eligible to play on this under-85-kg team?
34.
One type of BB gun uses a spring-driven plunger to blow the BB from its barrel. (a) Calculate the force constant of its plunger’s spring if you must compress it 0.150 m to drive the 0.0500-kg plunger to a top speed of 20.0 m/s. (b) What force must be exerted to compress the spring?
35.
When an 80.0-kg man stands on a pogo stick, the spring is compressed 0.120 m. (a) What is the force constant of the spring? (b) Will the spring be compressed more when he hops down the road?
36.
A spring has a length of 0.200 m when a 0.300-kg mass hangs from it, and a length of 0.750 m when a 1.95-kg mass hangs from it. (a) What is the force constant of the spring? (b) What is the unloaded length of the spring?
37.
The length of nylon rope from which a mountain climber is suspended has an effective force constant of 1.40×104N/m1.40×104N/m. (a) What is the frequency at which he bounces, given his mass plus and the mass of his equipment are 90.0 kg? (b) How much would this rope stretch to break the climber’s fall if he free-falls 2.00 m before the rope runs out of slack? (Hint: Use conservation of energy.) (c) Repeat both parts of this problem in the situation where twice this length of nylon rope is used.
15.3 Comparing Simple Harmonic Motion and Circular Motion
38.
The motion of a mass on a spring hung vertically, where the mass oscillates up and down, can also be modeled using the rotating disk. Instead of the lights being placed horizontally along the top and pointing down, place the lights vertically and have the lights shine on the side of the rotating disk. A shadow will be produced on a nearby wall, and will move up and down. Write the equations of motion for the shadow taking the position at t=0.0st=0.0s to be y=0.0my=0.0m with the mass moving in the positive y-direction.
39.
(a) A novelty clock has a 0.0100-kg-mass object bouncing on a spring that has a force constant of 1.25 N/m. What is the maximum velocity of the object if the object bounces 3.00 cm above and below its equilibrium position? (b) How many joules of kinetic energy does the object have at its maximum velocity?
40.
Reciprocating motion uses the rotation of a motor to produce linear motion up and down or back and forth. This is how a reciprocating saw operates, as shown below.
If the motor rotates at 60 Hz and has a radius of 3.0 cm, estimate the maximum speed of the saw blade as it moves up and down. This design is known as a scotch yoke.
41.
A student stands on the edge of a merry-go-round which rotates five times a minute and has a radius of two meters one evening as the sun is setting. The student produces a shadow on the nearby building. (a) Write an equation for the position of the shadow. (b) Write an equation for the velocity of the shadow.
15.4 Pendulums
42.
What is the length of a pendulum that has a period of 0.500 s?
43.
Some people think a pendulum with a period of 1.00 s can be driven with “mental energy” or psycho kinetically, because its period is the same as an average heartbeat. True or not, what is the length of such a pendulum?
44.
What is the period of a 1.00-m-long pendulum?
45.
How long does it take a child on a swing to complete one swing if her center of gravity is 4.00 m below the pivot?
46.
The pendulum on a cuckoo clock is 5.00-cm long. What is its frequency?
47.
Two parakeets sit on a swing with their combined CMs 10.0 cm below the pivot. At what frequency do they swing?
48.
(a) A pendulum that has a period of 3.00000 s and that is located where the acceleration due to gravity is 9.79m/s29.79m/s2 is moved to a location where the acceleration due to gravity is 9.82m/s29.82m/s2. What is its new period? (b) Explain why so many digits are needed in the value for the period, based on the relation between the period and the acceleration due to gravity.
49.
A pendulum with a period of 2.00000 s in one location (g=9.80m/s2g=9.80m/s2) is moved to a new location where the period is now 1.99796 s. What is the acceleration due to gravity at its new location?
50.
(a) What is the effect on the period of a pendulum if you double its length? (b) What is the effect on the period of a pendulum if you decrease its length by 5.00%?
15.5 Damped Oscillations
51.
The amplitude of a lightly damped oscillator decreases by 3.0%3.0% during each cycle. What percentage of the mechanical energy of the oscillator is lost in each cycle?
15.6 Forced Oscillations
52.
How much energy must the shock absorbers of a 1200-kg car dissipate in order to damp a bounce that initially has a velocity of 0.800 m/s at the equilibrium position? Assume the car returns to its original vertical position.
53.
If a car has a suspension system with a force constant of 5.00×104N/m5.00×104N/m, how much energy must the car’s shocks remove to dampen an oscillation starting with a maximum displacement of 0.0750 m?
54.
(a) How much will a spring that has a force constant of 40.0 N/m be stretched by an object with a mass of 0.500 kg when hung motionless from the spring? (b) Calculate the decrease in gravitational potential energy of the 0.500-kg object when it descends this distance. (c) Part of this gravitational energy goes into the spring. Calculate the energy stored in the spring by this stretch, and compare it with the gravitational potential energy. Explain where the rest of the energy might go.
55.
Suppose you have a 0.750-kg object on a horizontal surface connected to a spring that has a force constant of 150 N/m. There is simple friction between the object and surface with a static coefficient of friction μs=0.100μs=0.100. (a) How far can the spring be stretched without moving the mass? (b) If the object is set into oscillation with an amplitude twice the distance found in part (a), and the kinetic coefficient of friction is μk=0.0850μk=0.0850, what total distance does it travel before stopping? Assume it starts at the maximum amplitude.
Additional Problems
56.
Suppose you attach an object with mass m to a vertical spring originally at rest, and let it bounce up and down. You release the object from rest at the spring’s original rest length, the length of the spring in equilibrium, without the mass attached. The amplitude of the motion is the distance between the equilibrium position of the spring without the mass attached and the equilibrium position of the spring with the mass attached. (a) Show that the spring exerts an upward force of 2.00mg on the object at its lowest point. (b) If the spring has a force constant of 10.0 N/m, is hung horizontally, and the position of the free end of the spring is marked as y=0.00my=0.00m, where is the new equilibrium position if a 0.25-kg-mass object is hung from the spring? (c) If the spring has a force constant of 10.0 M/m and a 0.25-kg-mass object is set in motion as described, find the amplitude of the oscillations. (d) Find the maximum velocity.
57.
A diver on a diving board is undergoing SHM. Her mass is 55.0 kg and the period of her motion is 0.800 s. The next diver is a male whose period of simple harmonic oscillation is 1.05 s. What is his mass if the mass of the board is negligible?
58.
Suppose a diving board with no one on it bounces up and down in a SHM with a frequency of 4.00 Hz. The board has an effective mass of 10.0 kg. What is the frequency of the SHM of a 75.0-kg diver on the board?
59.
The device pictured in the following figure entertains infants while keeping them from wandering. The child bounces in a harness suspended from a door frame by a spring. (a) If the spring stretches 0.250 m while supporting an 8.0-kg child, what is its force constant? (b) What is the time for one complete bounce of this child? (c) What is the child’s maximum velocity if the amplitude of her bounce is 0.200 m?
60.
A mass is placed on a frictionless, horizontal table. A spring (k=100N/m)(k=100N/m), which can be stretched or compressed, is placed on the table. A 5.00-kg mass is attached to one end of the spring, the other end is anchored to the wall. The equilibrium position is marked at zero. A student moves the mass out to x=4.0cmx=4.0cm and releases it from rest. The mass oscillates in SHM. (a) Determine the equations of motion. (b) Find the position, velocity, and acceleration of the mass at time t=3.00s.t=3.00s.
61.
Find the ratio of the new/old periods of a pendulum if the pendulum were transported from Earth to the Moon, where the acceleration due to gravity is 1.63m/s21.63m/s2.
62.
At what rate will a pendulum clock run on the Moon, where the acceleration due to gravity is 1.63m/s21.63m/s2, if it keeps time accurately on Earth? That is, find the time (in hours) it takes the clock’s hour hand to make one revolution on the Moon.
63.
If a pendulum-driven clock gains 5.00 s/day, what fractional change in pendulum length must be made for it to keep perfect time?
64.
A 2.00-kg object hangs, at rest, on a 1.00-m-long string attached to the ceiling. A 100-g mass is fired with a speed of 20 m/s at the 2.00-kg mass, and the 100.00-g mass collides perfectly elastically with the 2.00-kg mass. Write an equation for the motion of the hanging mass after the collision. Assume air resistance is negligible.
65.
A 2.00-kg object hangs, at rest, on a 1.00-m-long string attached to the ceiling. A 100-g object is fired with a speed of 20 m/s at the 2.00-kg object, and the two objects collide and stick together in a totally inelastic collision. Write an equation for the motion of the system after the collision. Assume air resistance is negligible.
66.
Assume that a pendulum used to drive a grandfather clock has a length L0=1.00mL0=1.00m and a mass M at temperature T=20.00°C.T=20.00°C. It can be modeled as a physical pendulum as a rod oscillating around one end. By what percentage will the period change if the temperature increases by 10°C?10°C? Assume the length of the rod changes linearly with temperature, where L=L0(1+αΔT)L=L0(1+αΔT) and the rod is made of brass (α=18×10−6°C−1).(α=18×10−6°C−1).
67.
A 2.00-kg block lies at rest on a frictionless table. A spring, with a spring constant of 100 N/m is attached to the wall and to the block. A second block of 0.50 kg is placed on top of the first block. The 2.00-kg block is gently pulled to a position x=+Ax=+A and released from rest. There is a coefficient of friction of 0.45 between the two blocks. (a) What is the period of the oscillations? (b) What is the largest amplitude of motion that will allow the blocks to oscillate without the 0.50-kg block sliding off?
Challenge Problems
68.
A suspension bridge oscillates with an effective force constant of 1.00×108N/m1.00×108N/m. (a) How much energy is needed to make it oscillate with an amplitude of 0.100 m? (b) If soldiers march across the bridge with a cadence equal to the bridge’s natural frequency and impart 1.00×104J1.00×104J of energy each second, how long does it take for the bridge’s oscillations to go from 0.100 m to 0.500 m amplitude.
69.
Near the top of the Citigroup Center building in New York City, there is an object with mass of 4.00×105kg4.00×105kg on springs that have adjustable force constants. Its function is to dampen wind-driven oscillations of the building by oscillating at the same frequency as the building is being driven—the driving force is transferred to the object, which oscillates instead of the entire building. (a) What effective force constant should the springs have to make the object oscillate with a period of 2.00 s? (b) What energy is stored in the springs for a 2.00-m displacement from equilibrium?
70.
Parcels of air (small volumes of air) in a stable atmosphere (where the temperature increases with height) can oscillate up and down, due to the restoring force provided by the buoyancy of the air parcel. The frequency of the oscillations are a measure of the stability of the atmosphere. Assuming that the acceleration of an air parcel can be modeled as ∂2z′∂t2=gρo∂ρ(z)∂zz′∂2z′∂t2=gρo∂ρ(z)∂zz′, prove that z′=z0′et−N2√z′=z0′et−N2 is a solution, where N is known as the Brunt-Väisälä frequency. Note that in a stable atmosphere, the density decreases with height and parcel oscillates up and down.
71.
Consider the van der Waals potential U(r)=Uo[(Ror)12−2(Ror)6]U(r)=Uo[(Ror)12−2(Ror)6], used to model the potential energy function of two molecules, where the minimum potential is at r=Ror=Ro. Find the force as a function of r. Consider a small displacement r=Ro+r′r=Ro+r′ and use the binomial theorem:
(1+x)n=1+nx+n(n−1)2!x2+n(n−1)(n−2)3!x3+⋯(1+x)n=1+nx+n(n−1)2!x2+n(n−1)(n−2)3!x3+⋯,
to show that the force does approximate a Hooke’s law force.
72.
Suppose the length of a clock’s pendulum is changed by 1.000%, exactly at noon one day. What time will the clock read 24.00 hours later, assuming it the pendulum has kept perfect time before the change? Note that there are two answers, and perform the calculation to four-digit precision.
73.
(a) The springs of a pickup truck act like a single spring with a force constant of 1.30×105N/m1.30×105N/m. By how much will the truck be depressed by its maximum load of 1000 kg? (b) If the pickup truck has four identical springs, what is the force constant of each?