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Explore oscillatory motion, simple harmonic motion, and energy transfer in SHM through mathematical models and examples. Learn about resonance, damping, and periodicity.
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Vibrations and Waves Topics: Sample question: The gibbon will swing more rapidly and move more quickly through the trees if it raises its feet. How can we model the gibbon’s motion to understand this observation? • Equilibrium, restoring forces, and oscillation • Energy in oscillatory motion • Mathematical description of oscillatory motion • Damped oscillations • Resonance
Where in nature do we encounter vibrations? How are vibrations related to waves? What does it mean for motion to be periodic? When a vibration repeats itself, the motion is periodic. The simplest periodic motion is spring motion. Many systems resemble spring motion.
The object’s displacementvaries with time. Its maximumdisplacementis called its amplitude.
Simple Harmonic Motion Periodic Motion: When an object repeats the same motion over and over, we refer to the motion as periodic motion or oscillatory motion. A mechanical system that undergoes periodic motion always has a stable equilibrium position. The system will also always have a restoring forcethat acts to restore the system to its equilibrium position. Periodic motion under the action of a restoring force that is directly proportional to the displacement from equilibrium is called simple harmonic motion (SHM). By “directly proportional” we mean linear. In symbols, we have: Frestore x∝ Δx
Simple Harmonic Motion But the restoring force won’t do much restoring unless it opposes the motion away from equilibrium. So Frestore xand Δx need to have opposite signs. Putting this together we have the condition for SHM: Frestore x= Fx = – kΔx = – kx. This is Hooke’s Law. The constant k is called the spring constant. It is a measure of the stiffness of the spring. The SI units for k are N/m. In other SHM contexts, this can be generalized, and k will be replaced by an analogous quantity.
Simple Harmonic Motion simple harmonic motion (SHM) ↔ linear restoring force farther mass “runs away” ↔ more restoring force if mass goes right ↔ force points left if mass goes left ↔ force points right
SHM Period and Frequency The frequency of oscillation depends on physical properties of the oscillator; it does not depend on the amplitude of the oscillation.
Position, Velocity, and Acceleration Are Sinusoidal Functions of Time
Linear Restoring Forces and Simple Harmonic Motion If the restoring force is a linear function of the displacement from equilibrium, the oscillation is sinusoidal—simple harmonic motion. Slide 14-9
Sinusoidal Relationships Slide 14-10
Mathematical Description of Simple Harmonic Motion Slide 14-11
Energy in Simple Harmonic Motion As a mass on a spring goes through its cycle of oscillation, energy is transformed from potential to kinetic and back to potential. Slide 14-12
Solving Problems Slide 14-14
Checking Understanding A set of springs all have initial length 10 cm. Each spring now has a mass suspended from its end, and the different springs stretch as shown below. Now, each mass is pulled down by an additional 1 cm and released, so that it oscillates up and down. Rank the frequencies of the oscillating systems A, B, C and D, from highest to lowest. Slide 14-15
A series of pendulums with different length strings and different masses is shown below. Each pendulum is pulled to the side by the same (small) angle, the pendulums are released, and they begin to swing from side to side. Rank the frequencies of the five pendulums, from highest to lowest. Slide 14-16
A ball on a spring is pulled down and then released. Its subsequent motion appears as follows: • At which of the above times is the displacement zero? • At which of the above times is the velocity zero? • At which of the above times is the acceleration zero? • At which of the above times is the kinetic energy a maximum? • At which of the above times is the potential energy a maximum? • At which of the above times is kinetic energy being transformed to potential energy? • At which of the above times is potential energy being transformed to kinetic energy? Slide 14-17
A pendulum is pulled to the side and released. Its subsequent motion appears as follows: • At which of the above times is the displacement zero? • At which of the above times is the velocity zero? • At which of the above times is the acceleration zero? • At which of the above times is the kinetic energy a maximum? • At which of the above times is the potential energy a maximum? • At which of the above times is kinetic energy being transformed to potential energy? • At which of the above times is potential energy being transformed to kinetic energy? Slide 14-18
Examples The first astronauts to visit Mars are each allowed to take along some personal items to remind them of home. One astronaut takes along a grandfather clock, which, on earth, has a pendulum that takes 1 second per swing, each swing corresponding to one tick of the clock. When the clock is set up on Mars, will it run fast or slow? A 5.0 kg mass is suspended from a spring. Pulling the mass down by an additional 10 cm takes a force of 20 N. If the mass is then released, it will rise up and then come back down. How long will it take for the mass to return to its starting point 10 cm below its equilibrium position? Slide 14-19
Example We think of butterflies and moths as gently fluttering their wings, but this is not always the case. Tomato hornworms turn into remarkable moths called hawkmoths whose flight resembles that of a hummingbird. To a good approximation, the wings move with simple harmonic motion with a very high frequency—about 26 Hz, a high enough frequency to generate an audible tone. The tips of the wings move up and down by about 5.0 cm from their central position during one cycle. Given these numbers, • What is the maximum velocity of the tip of a hawkmoth wing? • What is the maximum acceleration of the tip of a hawkmoth wing? Slide 14-20
Example The deflection of the end of a diving board produces a linear restoring force, as we saw in Chapter 8. A diving board dips by 15 cm when a 65 kg person stands on its end. Now, this person jumps and lands on the end of the board, depressing the end by another 10 cm, after which they move up and down with the oscillations of the end of the board. • Treating the person on the end of the diving board as a mass on a spring, what is the spring constant? • For a 65 kg diver, what will be the oscillation period? • For the noted oscillation, what will be the maximum speed? • What amplitude would lead to an acceleration greater than that of gravity—meaning the person would leave the board at some point during the cycle? Slide 14-21
Example In Chapter 10, we saw that the Achilles tendon will stretch and then rebound, storing and returning energy during a step. We can model this motion as that of a mass on a spring. It’s far from a perfect model, but it does give some insight. Suppose a 60 kg person stands on a low wall with her full weight on the balls of one foot and the heel free to move. The stretch of the Achilles tendon will cause her center of mass to lower by about 2.5 mm. • What is the value of k for this system? • Given the mass and the spring constant, what would you expect for the period of this system were it to undergo an oscillation? • When the balls of the feet take the weight of a stride, the tendon spring begins to stretch as the body moves down; kinetic energy is being converted into elastic potential energy. Ideally, when the foot is leaving the ground, the cycle of the motion will have advanced so that potential energy is being converted to kinetic energy. What fraction of an oscillation period should the time between landing and lift off correspond to? Given the period you calculated above, what is this time? • Sprinters running a short race keep their foot in contact with the ground for about 0.10 s, some of which corresponds to the heel strike and subsequent rolling forward of the foot. Given this, does the final number you have calculated above make sense? Slide 14-22
Example A 204 g block is suspended from a vertical spring, causing the spring to stretch by 20 cm. The block is then pulled down an additional 10 cm and released. What is the speed of the block when it is 5.0 cm above the equilibrium position? Slide 14-23
Damping A 500 g mass on a string oscillates as a pendulum. The pendulum’s energy decays to 50% of its initial value in 30 s. What is the value of the damping constant? Slide 14-24
Resonance Slide 14-25
Four different masses are hung from four springs with unstretched length 10 cm, causing the springs to stretch as noted in the following diagram: Now, each of the masses is lifted a small distance, released, and allowed to oscillate. Rank the oscillation frequencies, from highest to lowest. Slide 14-26
Four 100 g masses are hung from four springs, each with unstretched length 10 cm. The four springs stretch as noted in the following diagram: Now, each of the masses is lifted a small distance, released, and allowed to oscillate. Rank the oscillation frequencies, from highest to lowest. Slide 14-27
A pendulum is pulled to the side and released. Rank the following positions in terms of the speed, from highest to lowest. There may be ties. Slide 14-28
A typical earthquake produces vertical oscillations of the earth. Suppose a particular quake oscillates the ground at a frequency 0.15 Hz. As the earth moves up and down, what time elapses between the highest point of the motion and the lowest point? • 1 s • 3.3 s • 6.7 s • 13 s Slide 14-29
Answer A typical earthquake produces vertical oscillations of the earth. Suppose a particular quake oscillates the ground at a frequency 0.15 Hz. As the earth moves up and down, what time elapses between the highest point of the motion and the lowest point? • 3.3 s Slide 14-30
Additional Example Walter has a summer job babysitting a 18 kg youngster. He takes his young charge to the playground, where the boy immediately runs to the swings. The seat of the swing the boy chooses hangs down 2.5 m below the top bar. “Push me,” the boy shouts, and Walter obliges. He gives the boy one small shove for each period of the swing, in order keep him going. Walter earns $6 per hour. While pushing, he has time for his mind to wander, so he decides to compute how much he is paid per push. How much does Walter earn for each push of the swing? Slide 14-31
Additional Examples A 500 g block is attached to a spring on a frictionless horizontal surface. The block is pulled to stretch the spring by 10 cm, then gently released. A short time later, as the block passes through the equilibrium position, its speed is 1.0 m/s. What is the block’s period of oscillation? What is the block’s speed at the point where the spring is compressed by 5.0 cm? A mass bounces up and down on a spring. The oscillation decays with a time constant of 50 s. If the oscillation begins with an amplitude of 20 cm, how long will it take until the amplitude has decreased by half to 10 cm? If the oscillation begins with an amplitude of 20 cm, how long will it take until the energy of the oscillation has decreased by half? Slide 14-32
Reading Quiz • The type of function that describes simple harmonic motion is • linear • exponential • quadratic • sinusoidal • inverse Slide 14-2
Answer • The type of function that describes simple harmonic motion is • sinusoidal Slide 14-3
Reading Quiz • A mass is bobbing up and down on a spring. If you increase the amplitude of the motion, how does this affect the time for one oscillation? • The time increases. • The time decreases. • The time does not change. Slide 14-4
Answer • A mass is bobbing up and down on a spring. If you increase the amplitude of the motion, how does this affect the time for one oscillation? • The time does not change. Slide 14-5
