Physics 101: Lecture 20 Elasticity and Oscillations
Young’s Modulus Spring F = -k x What happens to “k” if cut spring in half? A) decreases B) same C) increases k is inversely proportional to length! Define Strain = DL / L المطاوعــة Stress = F/A الضغط Now Stress = Y Strain F/A = Y DL/L k = Y A/L from F = k x Or F = k DL Y (Young’s Modulus) independent of L 09
Review Energy in SHM A mass is attached to a spring and set to motion. The maximum displacement is x=A Energy = U + K = constant! = ½ k x2 + ½ m v2 At maximum displacement x=A, v = 0 Energy = ½ k A2 + 0 At zero displacement x = 0 Energy = 0 + ½ mvm2 ½ k A2 = ½ m vm2 vm = (k/m) A Analogy with gravity/ball x PES m x x=0 05
Kinetic Energy ACT ½kA2 = ½mvm2 In Case 1 a mass on a spring oscillates back and forth. In Case 2, the mass is doubled but the spring and the amplitude of the oscillation are the same as in Case 1. In which case is the maximum kinetic energy of the mass bigger? A. Case 1 B. Case 2 C. Same ½kA2 = ½mvm2 09
Potential Energy ACT In Case 1 a mass on a spring oscillates back and forth. In Case 2, the mass is doubled but the spring and the amplitude of the oscillation are the same as in Case 1. In which case is the maximum potential energy of the mass and spring bigger? A. Case 1 B. Case 2 C. Same Maximum displacement x = A Energy = ½ k A2 + 0 Same for both! 12
Velocity ACT In Case 1 a mass on a spring oscillates back and forth. In Case 2, the mass is doubled but the spring and the amplitude of the oscillation are the same as in Case 1. Which case has the larger maximum velocity? 1. Case 1 2. Case 2 3. Same Same maximum Kinetic Energy K = ½ m v2 smaller mass requires larger v 16
Simple Harmonic Motion: Quick Review x(t) = [A]cos(t) v(t) = -[A]sin(t) a(t) = -[A2]cos(t) x(t) = [A]sin(t) v(t) = [A]cos(t) a(t) = -[A2]sin(t) OR xmax = A vmax = A amax = A2 Period = T (seconds per cycle) Frequency = f = 1/T (cycles per second) Angular frequency = = 2f = 2/T 19
Natural Period T of a Spring Simple Harmonic Oscillator w = 2 p f = 2 p / T x(t) = [A] cos(wt) v(t) = -[Aw] sin(wt) a(t) = -[Aw2] cos(wt) write F=ma -k x = m a -k A = m amax -k A = m (-A w2) Aw2 = (k/m) A w = (k/m) A,m,k dependence demo 23
Period ACT If the amplitude of the oscillation (same block and same spring) is doubled, how would the period of the oscillation change? (The period is the time it takes to make one complete oscillation) A. The period of the oscillation would double. B. The period of the oscillation would be halved C. The period of the oscillation would stay the same x +2A t -2A 27
Equilibrium position and gravity If we include gravity, there are two forces acting on mass. With mass, new equilibrium position has spring stretched d SFy = 0 kd – mg = 0 d = mg/k Let this point be y=0 SF = ma k(d-y) – mg = ma -k y = ma Same as horizontal! SHO New equilibrium position y=0 corresponds to height -d 31
Vertical Spring ACT Two springs with the same k but different equilibrium positions are stretched the same distance d and then released. Which would have the larger maximum kinetic energy? 1) M 2) 2M 3) Same PE = 1/2k y2 PE = 1/2k y2 Y=0 Just before being released, v=0 y=d Etot = 0 + ½ k d2 Same total energy for both When pass through equilibrium all of this energy will be kinetic energy again - same for both! Y=0 33
Pendulum Motion For small angles T = mg cos(q) = mg Bowling ball pendulum For small angles T = mg cos(q) = mg Tx = -mg sin(q) = -mg x/L Note: F proportional to x! S Fx = m ax -mg (x/L) = m ax ax = -(g/L) x Recall for SHO a = - w2 x w = (g/L) T = 2 p (L/g) Period does not depend on A, or m! q L T m x mg 37
Preflight 1 Suppose a grandfather clock (a simple pendulum) runs slow. In order to make it run on time you should: 1. Make the pendulum shorter 2. Make the pendulum longer CORRECT 38
Elevator ACT A pendulum is hanging vertically from the ceiling of an elevator. Initially the elevator is at rest and the period of the pendulum is T. Now the pendulum accelerates upward. The period of the pendulum will now be A. greater than T B. equal to T C. less than T g is effectively bigger, T is lower. 42
Preflight Imagine you have been kidnapped by space invaders and are being held prisoner in a room with no windows. All you have is a cheap digital wristwatch and a pair of shoes (including shoelaces of known length). Explain how you might figure out whether this room is on the earth or on the moon “Since you know the length of the shoelace then you can build a pendulum and see how long it takes for one period. If its longer than what you expect on earth than you must be on the moon, if its the same you're on earth.” 50
Alien Preflight “You could drop a shoe from a known height and also measure the time it takes for the shoe to hit the ground. Then you can use x=xo + vot + 1/2at2 to find the acceleration. The shoe starts at rest (vo=0). If the acceleration is less than 9.8, you're on the moon.” “I could jump. that would be enough really. The gravitational difference is quite significant between the two. but I suppose I could make a pendulum of the watch and shoelace if I wanted to do extra work and was bored.” “I have absolutely no idea. I have been studying for the exam tonight since last sunday, and all day today. I came home from the exam only to slave over the physics homework until now, and now I'm doing this preflight. I am on physics OVERLOAD and going cross eyed. I seriously think that it is cruel and unfair that we have a homework due after a test. As for this pendulum, I think that you could tie a shoe to a shoelace and start it swinging in a pendulum motion. The period of the pendulum (T) could be approximated by 2*Pi *square root (length of shoelaces/g). You have your watch to time the pendulum, and so you could determine if g is the acceleration of the Earth (9.8) or a martian planet!” 46
Alien Preflight “First you tie both shoe laces together and attach that to the shoe. this will give you a good amount of torque because the radius is longer. then you pretend to have a heart attack. After a while, the aliens will most likely come and check on you. once they do, get up, start swinging the shoe lace/shoe combo and hit one of them in the face. if there is another one, throw your other shoe at him. if there are even more, headbutt them into oblivion. continue headbutting into oblivion until all the aliens are down. find the main control center and pilot the space ship to the nearest pawn shop and sell the space ship for one gazillion dollars and 33 cents. get 33 cents in pennies and start throwing them at the aliens until they tell you were they took you. the end. or since small oscillations depend on gravity, you can tie the shoe laces to the shoe. then figure out what 2*pi*sqrt(L/9.8) is and then swing the 'pendulum' and see if the period for that is the same.” 46
Summary Simple Harmonic Motion Springs Pendulum (Small oscillations) Occurs when have linear restoring force F= -kx x(t) = [A] cos(wt) v(t) = -[Aw] sin(wt) a(t) = -[Aw2] cos(wt) Springs F = -kx U = ½ k x2 w = sqrt(k/m) Pendulum (Small oscillations) w = (g/L) *incorrect in lecture notes 50