103 Discussion 15 Solution - SHM

Oscillations – Group Sheet Physics 103 – General Physics 1 Ç√ Ç√ Ç√ Ç√ Ç Ç 1) Circle the correct words in the following statements: To increase the frequency of a mass/spring system, use a stronger / weaker spring. To increase the frequency of a mass/spring system, use a lighter / heavier mass. To decrease the frequency of a mass/spring system, use a stronger / weaker spring. To decrease the frequency of a mass/spring system, use a lighter / heavier mass. 2) A duck is bobbing up and down on Lake Monona. You can model this by imagining the duck is oscillating at the end of a spring. If we graph the duck’s position over time we get a plot like the one below. (Note: This is a sine curve.) Which statement describes the duck’s motion at point P? a) Positive velocity and positive acceleration b) Positive velocity and negative acceleration c) Positive velocity and zero acceleration a) Negative velocity and positive acceleration b) Negative velocity and negative acceleration c) Negative velocity and zero acceleration a) Zero velocity and positive acceleration b) Zero velocity and negative acceleration c) Zero velocity and zero acceleration 3) You’d like to increase the period of a simple pendulum by a factor of 3. What can you do? Select all that apply. a. Increase the length of the string by 3 times. b. Increase the mass of the object on the string by √3 times. c. Increase the initial angle of deflection by 3 times. d. Increase the length of the string by 9 times. e. Decrease the mass of the object on the string by 3 times. 4) Name one instance when someone in your group experienced the phenomenon of resonance and describe it here. Be sure to check in with to make sure this was an actual case of resonance. Your group member names: Here, the duck is moving up (+) but its acceleration always points back toward equilibrium, so a is (–). We also know v and a have to be in opposite directions because the duck is slowing down.)

Oscillations Physics 103 – General Physics 2 (250 cm/s) x (1 m/100 cm) = 2.5 m/s which we can use in: K i = (1/2)mv 2 = (1/2)(4 kg)(2.5 m/s) 2 = 12.5 J U i = (1/2)kx 2 = (1/2)(2500 N/m)(0.03 m) 2 = 1.125 J E tot = U + K = 1.125 J + 12.5 J = 13.625 J At this point, all the energy in the system is potential, so U = E tot = 13.625 J Useful Expressions: F spring = -kΔx U spring = ! " kx 2 E tot = ! " kA 2 = ! " mv max 2 = ! " kx 2 + ! " mv 2 ω 0 = ! # $ f = ! "% ! # $ T = 2π ! $ # = 1/f for a spring-mass system ω 0 = ! & ’ f = ! "% ! & ’ T = 2π ! ’ & = 1/f for a pendulum x = Acos(ω 0 t) v = –ω 0 Asin(ω 0 t) a = –ω 0 2 Acos(ω 0 t) if x = 0 at t = 0 A(ω) = ! ! "# " $% ! " &% " ’() " % " for an oscillator driven with force F 0 5) You have a 4 kg mass on the end of a spring with a spring constant k = 2500 N/m. The mass is in its equilibrium position. Suppose you then compress the spring by 3 cm and give it a kick so that it has an initial velocity toward the base of the spring equal to 250 cm/s. What is the amplitude of the resulting oscillation? A) We’ll take this one step at a time. What is the kinetic energy of the mass just after you kick it? Assume it hasn’t yet moved. B) What is the potential energy of the mass/spring at this same moment? C) So what is the total mechanical energy of the system? D) Now imagine that the mass has moved a distance from the equilibrium point equal to the amplitude of the oscillation. What is the potential energy of the mass/spring system at this moment?

Oscillations Physics 103 – General Physics 4 The mass and the amplitude of the oscillation. These assume a small θ of course, which is the same assumption we make whenever we deal with a simple pendulum. f )//0 f 1,23+ & = ! "4 ! g )//0 L & ! "4 ! g 1,23+ L & 0 = 1g )//0 1g 1,23+ 2 = 0.4 Since the frequency of a pendulum is proportional to the square root of g, if g does down the frequency should decrease. There would be no frequency. Without gravity, there’s no restoring force and therefore no oscillation. The mass would just sit wherever you put it. We could give it a larger amplitude (more initial PE since gravitational potential energy increases with height and a larger initial angle of displacement means the mass starts a little higher) or give it an extra push (more initial KE). C) What could you change about the pendulum and not affect the frequency? D) The acceleration due to gravity on the surface of the moon is 1.6 m/s 2 . Would the frequency of the pendulum increase, decrease, or remain the same if it were transported to the moon? E) What is the ratio of the pendulum’s frequency on the moon to its frequency on the Earth? F) What would the frequency of the pendulum be if you put it on the International Space Station in zero-g conditions? G) Like all oscillators, the pendulum has some total amount of mechanical energy. What could you change that would increase this energy? (Hint: Think about the things that affect the total mechanical energy of a mass- spring system and find the analogues to those things with a pendulum. What determined the kinetic energy of the pendulum? What is the potential energy due to and what affects it?)

Oscillations Physics 103 – General Physics 5 ω = ! ࠵? ࠵? $ = ! 2100 N/m 6 kg $ = 18.7 rad/s x = Acos(ωt) = (20 cm)cos((18.7 rad/s)(2.5 s)) = –18.6 cm v = –Aωsin(ωt) = –(20 cm)(18.7 rad/s)sin((18.7 rad/s)(2.5 s)) = –137 cm/s = -1.37 m/s a = –Aω 2 cos(ωt) = –(20 cm)(18.7 rad/s) 2 cos((18.7 rad/s)(2.5 s)) = 6511 cm/s 2 = 65.1 m/s 2 7) The sine and cosine equations for x, v, and a contain much more information about an oscillating mass/spring system than you might think. To see that, imagine the following: You have a mass/spring system with m = 6 kg and k = 2100 N/m. You set up an oscillation that has an amplitude of 20 cm. Ignore any additional phase angle, ϕ. A) What is the position of the mass after 2.5 s? (Remember to use the correct angular units on your calculator when finding the cosine here.) B) What is the velocity of the mass after 2.5 s? C) What is the acceleration of the mass after 2.5 s? D) Think about your answers to A, B, and C and draw a picture of the mass below at t = 2.5 s. (Assume + means toward the right. The dashed line is the equilibrium position.) Indicate the position of the mass and draw the velocity and acceleration vectors. E) Sum up your picture by choosing the right words in the statements below: The mass is to the right / left of the equilibrium position. The mass is moving toward / away from the equilibrium position. The mass is speeding up / slowing down. The object is to the left of the equilibrium position, so the acceleration is pointing right (toward equilibrium) but it hasn’t yet reached it’s max amplitude, so it’s still moving left. Because the velocity and acceleration are in opposite directions.

Oscillations Physics 103 – General Physics 7