Simple Harmonic Motion & Waves

Vocabulary Period: the time for one cycle of simple harmonic motion, or the time for a full wavelength to pass a position. Frequency: the number of cycles, or the number of wavelengths passing a position, in one second. The unit for frequency is Hertz which means per second.

Vocabulary Contd. Amplitude: the distance from the midpoint of a wave to its crest or trough or the distance from the midpoint of simple harmonic motion to the maximum displacement Wavelength: measured from peak to peak or between any successive identical points on a wave.

Vocabulary Contd. Spring Constant: k, measured in units of newtons per meter (N/m), is related to the stiffness of the spring Restoring force: is any force that always pushes an object toward an equilibrium position.

Vocabulary Contd. Nodes: are the stationary points on a standing wave. Antinodes: are the positions on standing waves with the largest amplitudes.

Simple Harmonic Motion (SHM) Refers to the back-and-forth oscillation whose position-time graph looks like a sine function Examples: mass vibrating on a spring and a pendulum

Period of a mass on a spring in SHM The bigger the mass means a larger or longer period of oscillation and doubling the mass would multiply the period by the square root of 2. Note: The AP exam probably ask something along the lines of if you double the mass what affect does that have on the period

What about Amplitude? What happens to the period if you double the amplitude? Trick question, amplitude doesn’t affect the period.

Frequency vs. Period Frequency and period are inverses of one another

Restoring Force The amount of restoring force exerted by a spring is given as Hooke’s Law

Restoring Force The force of the spring is greatest when the spring is most stretched, but zero at the equilibrium positon. Acceleration changes (max at the endpoints to zero in the middle), you cannot use kinematics for a spring (kinematics is used for constant acceleration ONLY) SO when looking for the speed, you must use the conservation of energy

Potential Energy of a Spring Spring potential energy

Energy The energy stored in the spring is largest at the endpoints and zero at the equilibrium position. Spring energy is completely converted into kinetic energy at equilibrium. So where is the cart’s speed the greatest? Largest kinetic energy is the largest speed so at the equilibrium

Pendulum Pendulum can be treated the same as a spring. It’s still in harmonic motion and requires an energy approach not kinematics to determine speed at any position.

Period of a Pendulum The period of a pendulum Note: Again, the AP exam will probably ask something along the lines of asking you to determine the order from greatest to least of a particular variable.

Ranking 1 Start A B C D Rank the lettered positions from greatest to least by the bob’s gravitational potential energy. Gravitational potential energy is mgh. Bob always has the same mass and g can’t change so it’s determined by height. So the highest vertical height has the greatest GPE. D>C=A>B

Ranking 2 Start A B C D Rank the lettered positions from greatest to least by the bob’s total mechanical energy. No friction or nonconservative forces acting on it and no internal structure to allow for internal energy, the total mechanical energy doesn’t change. A=B=C=D

Ranking 3 Start A B C D Rank the lettered positions from greatest to least by the bob’s speed Since GPE is converted to KE the bob moves fastest when the GPE is the smallest. B>C=A>D

Challenge Question The gravitational field at the surface of Jupiter is 26 n/kg and on the surface of the Moon, 1.6 N/kg. Rank this pendulum’s period near these two planets and earth. What about the ranking of the frequency? Since g is in the denominator of the period equation, the lowest gravitational field will have the greatest period. Tmoon>Tearth>Tjupiter. The ranking of the frequency would be opposite because frequency is the inverse of period. A bigger g leads to a smaller period but a bigger frequency.

Mechanical Waves The AP exam focuses on Mechanical waves like sound or waves on the surface of an ocean, not light waves (electromagnetic)

Transverse Waves Whenever the motion of a material is at right angles to the direction in which the wave travels, the wave is a transverse wave.

Energy The energy carried by the wave depends on the wave’s amplitude. Draw a wave the with low energy and one with high energy.

Frequency When a wave changes materials, its frequency remains the same. However, the speed and wavelength may increase. Multiplying by the same frequency means the wavelength has to get bigger for a bigger value of v. The wave will look wider than in the new spring.

Longitudinal Waves When a material vibrates parallel to the direction of the wave, the wave is a longitudinal wave.

Wavelength Try drawing a longitudinal wave and labeling a wavelength. A wavelength is defined as the distance between two identical positions on the wave.

Interference When two waves collide, they don’t bounce or stick like objects do. Rather, the waves interfere. They form one single wave for just a moment, and then the waves continue on their merry way.

Crest of one wave overlaps the crest of another, result is a wave of increased amplitude.

The crest of one wave overlaps with the trough of another, the result is a wave of reduced or no amplitude.

Standing Waves Is a wave that appears to stay in one place. The reason a standing wave exists revolves around interference. Waves are traveling back and forth on the string, reflecting off of each end and interfering with each other all willy-nilly. The net effect of all this interference is a pattern of nodes and antinodes.

Standing Waves Generally, when you pluck a string, you produce a standing wave of the longest possible wavelength and thus the smallest possible frequency.

Wavelength The wavelength of a standing wave is twice the node-to-node distance.

Frequency For a string fixed at both ends, or for a pipe open at both ends, the smallest frequency of standing waves is given by V is the speed of the waves on the string or in the pipe, when dealing with an open pipe the speed is generally air 340 m/s. Other harmonic frequencies for this string or pipe must be whole-number multiples of the fundamental frequency. If the fundamental frequency is 100 Hz then it can play 200 Hz, 300 Hz, 400 Hz.

Pitch & Loudness The loudness of a note depends on the sound wave’s amplitude. The pitch of a musical note depends on the sound wave’s frequency

Frequency For a pipe closed at one end, or for a string fixed at one end but free at the other, the smallest frequency of standing waves is given by Other harmonic frequencies for this string or pipe must be odd multiples of the fundamental frequency. So if the fundamental frequency is 85 Hz, then the other frequencies are 255 Hz, 425 Hz etc…. 4

Beats Beats are rhythmic interference that occurs when two notes of unequal but close frequencies are played.

Doppler Effect The Doppler Effect is the apparent change in a wave’s frequency that you observe whenever the source of the wave is moving toward or away from you.