Classical Mechanics Lecture 21 Today s Concept: Simple Harmonic Mo7on: Mass on a Spring Mechanics Lecture 21, Slide 1
The Mechanical Universe, Episode 20: Harmonic Motion http://www.learner.org/vod/login.html?pid=565
Your Comments Good stuff :) Easier than angular stuff. Just when we thought it couldn't get any worse... I did not understand whether to use sine or cosine to find the displacement. This stuff looks much more easier to look at then the last unit. Is there such a thing as Rota7onal Harmonic Mo7on? There bemer not be... Go over adding 2 graphs together. How it would look like aoer adding or subtrac7ng. is this part of the main course or is this like appendix? Not really from this lecture, but you might want to explain the difference between frequency, angular frequency, and angular velocity. It would be much easier to understand if Morgan Freeman was the narrator. I dont like cross product. Out of everything we've learned so far in physics, this was my hands down favourite unit. :D Aside from torque and precession. how can we determine phi at the start?. How does physics help bring me closer to discovering the meaning of life? Mechanics Lecture 21, Slide 2
How does physics help bring me closer to discovering the meaning of life? The answer to the ques7on of life, the universe, and everything: 42 Next 7me I will explain why! Mechanics Lecture 21, Slide 3
CheckPoint k m A 0 A x A mass on a spring moves with simple harmonic mo7on as shown. Where is the accelera7on of the mass most posi7ve? A) x = A B) x = 0 C) x = A Mechanics Lecture 21, Slide 4
k m A mass on a spring moves with simple harmonic mo7on as shown. Where is the accelera7on of the mass most posi7ve? A) x = A B) x = 0 C) x = A A 0 A x A) The posi7ve force on the mass is greatest when the spring most compressed. B) Because at x=a, - A there is no accelera7on. C) Force is greatest at this point so therefore accelera7on is too. Mechanics Lecture 21, Slide 5
Most general solu7on: x(t) = Acos(ωt φ) Mechanics Lecture 21, Slide 6
demo Mechanics Lecture 21, Slide 7
SHM Dynamics What does angular frequency ω have to do with moving back and forth in a straight line? y = R cosθ = R cos(ωt) 5 4 y 3 θ 2 1 x 1 0 1 2 3 7 8 1 θ 6 7 8 1 4 5 6 Mechanics Lecture 21, Slide 8
SHM Solution Drawing of Acos(ωt) T = 2π /ω A 2π π A π 2π θ Mechanics Lecture 21, Slide 9
SHM Solution Drawing of Acos(ωt φ) φ 2π π π θ 2π Mechanics Lecture 21, Slide 10
SHM Solution Drawing of Acos(ωt π /2) = Asin(ωt) π /2 2π π π 2π θ Mechanics Lecture 21, Slide 11
In the slide 7tled, "example", the decision to choose the equa7on "A*cos(w*t +'phi') seemed like an arbitrary decision. Please explain why that equa7on was chosen over the other general solu7on equa7ons. The descrip7on of the phase angle is a limle confusing too. φ 2π π π θ 2π Drawing of Acos(ωt φ) Mechanics Lecture 21, Slide 12
In the slide 7tled, "example", the decision to choose the equa7on "A*cos(w*t +'phi') seemed like an arbitrary decision. Please explain why that equa7on was chosen over the other general solu7on equa7ons. The descrip7on of the phase angle is a limle confusing too. Is this a sine or a cosine? Mechanics Lecture 21, Slide 13
CheckPoint Suppose the two sinusoidal curves shown above are added together. Which of the plots shown below best represents the result? A) B) C) Mechanics Lecture 21, Slide 14
A) B) C) A) The SUM of them would be a limle irregular. B) The sum of two waves with equal frequency gives another signal with the same frequency and different amplitude/phase. C) When adding the two curves together the aplitude decreases and the frequency increases. Mechanics Lecture 21, Slide 15
Any linear combina7on of sines and cosines having the same frequency will result in a sinusoidal curve with the same frequency. Mechanics Lecture 21, Slide 16
Can we talk more about the trigonometrical func7ons? Like in ques7on #2. Suppose a = ωt + α b = ωt + β sin(ωt + α) + cos(ωt + β) = 2 sin ωt + α + β α β cos 2 2 Mechanics Lecture 21, Slide 17
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CheckPoint Case 1 k m k 1 0 1 x m Case 2 2 0 2 x In the two cases shown the mass and the spring are iden7cal but the amplitude of the simple harmonic mo7on is twice as big in Case 2 as in Case 1. How are the maximum veloci7es in the two cases related: A) v max,2 = v max,1 B) v max,2 = 2v max,1 C) v max,2 = 4v max,1 Mechanics Lecture 21, Slide 20
Case 1 k m k 2 0 2 x 1 0 1 x m Case 2 How are the maximum veloci7es in the two cases related: A) v max,2 = v max,1 B) v max,2 = 2v max,1 C) v max,2 = 4v max,1 A) The velocity does not depend on the amplitude. B) Vmax=ω*D so twice the amplitude means twice the velocity C) KE will depend on spring energy, which has an (x)^2 term. So doubling that x term will quadruple the KE. Mechanics Lecture 21, Slide 21
Clicker Question A mass oscillates up & down on a spring. Its posi7on as a func7on of 7me is shown below. At which of the points shown does the mass have posi7ve velocity and nega7ve accelera7on? y(t) (A) (C) t (B) Mechanics Lecture 21, Slide 22
The slope of y(t) tells us the sign of the velocity since y(t) and a(t) have the opposite sign since a(t) = ω 2 y(t) a < 0 v < 0 y(t) a < 0 v > 0 (A) (B) a > 0 v > 0 (C) t The answer is (C). Mechanics Lecture 21, Slide 23
Clicker Question A mass hanging from a ver7cal spring is lioed a distance d above equilibrium and released at t = 0. Which of the following describes its velocity and accelera7on as a func7on of 7me? A) v(t) = v max sin(ωt) a(t) = a max cos(ωt) B) v(t) = v max sin(ωt) a(t) = a max cos(ωt) k y C) v(t) = v max cos(ωt) a(t) = a max cos(ωt) t = 0 m d 0 (both v max and a max are posi7ve numbers) Mechanics Lecture 21, Slide 24
Clicker Question Since we start with the maximum possible displacement at t = 0 we know that: y = d cos(ωt) t = 0 k m y d 0 Mechanics Lecture 21, Slide 25
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At t = 0, y = 0, moving down Use energy conserva7on to find A Mechanics Lecture 21, Slide 27
Or similarly Mechanics Lecture 21, Slide 28
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