Today’s Concept: Simple Harmonic Motion: Mass on a Spring Physics 211 Lecture 21 Today’s Concept: Simple Harmonic Motion: Mass on a Spring

3rd Exam is next Wednesday at 7pm: Covers lectures 14-20 (not the stuff we are doing today) Sign up for the conflict if you need to before the end of this week. As before, email Dr. Jain if you have a double conflict (rdjain1@illinois.edu). Next Tuesdays class will be a review (Fall 2010 exam) . Don’t forget about office hours !!

CheckPoint k -A A x m A mass on a spring moves with simple harmonic motion as shown. Where is the acceleration of the mass most positive? A) x = -A B) x = 0 C) x = A

k -A A x m A mass on a spring moves with simple harmonic motion as shown. Where is the acceleration of the mass most positive? A) x = -A B) x = 0 C) x = A A) At x= -A force is maximized in positive direction, so acceleration is as well.. B) When the kinetic energy is at it's max, so is it's velocity and acceleration

Most general solution: x(t) = Acos(wt-f)

The depiction of how the oscillation resembled circular motion was very interesting. demo

SHM Solution Drawing of Acos(t) T = 2 / A  -    A

SHM Solution Drawing of Acos(t - f) f    - 

SHM Solution Drawing of Acos(t - p /2) = Asin(t) Similarly Acos(t) = Asin(t + p /2) p /2  -   

x(t) = Acos(wt-f) = Asin(wt-f+p/2 ) = Asin(wt-f’) In the slide titled, "example", the decision to choose the equation "A*cos(w*t+'phi') seemed like an arbitrary decision. Please explain why that equation was chosen over the other general solution equations. The description of the phase angle is a little confusing too. x(t) = Acos(wt-f) = Asin(wt-f+p/2 ) f = Asin(wt-f’)    -  Drawing of Acos(t - f)

Can we talk more about the trigonometrical functions Can we talk more about the trigonometrical functions? Like in question #2. a = wt + a Suppose b = wt + b

CheckPoint Suppose the two sinusoidal curves shown above are added together. Which of the plots shown below best represents the result? you just look at the max and min points A) sum of 2 sinusoidals of one frequency has the same frequency. B) when 2 sinusoidal curves are plotted together, the frequency increases C)

Any linear combination of sines and cosines having the same frequency will result in a sinusoidal curve with the same frequency.

CheckPoint Case 1 k -2 2 x m Case 2 -1 1 In the two cases shown the mass and the spring are identical but the amplitude of the simple harmonic motion is twice as big in Case 2 as in Case 1. How are the maximum velocities in the two cases related: A) vmax,2 = vmax,1 B) vmax,2 = 2vmax,1 C) vmax,2 = 4vmax,1

Case 1 k -2 2 x m Case 2 -1 1 How are the maximum velocities in the two cases related: A) vmax,2 = vmax,1 B) vmax,2 = 2vmax,1 C) vmax,2 = 4vmax,1 A) Velocity is not related to the amplitude but to the mass and the spring constant thus these are the same in this case. B) Vmax=w*D double the amplitude=twice the velocity C) usually when something is doubled, the other ratio is 4:1

Clicker Question A mass oscillates up & down on a spring. Its position as a function of time is shown below. At which of the points shown does the mass have positive velocity and negative acceleration? t y(t) (A) (B) (C)

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) = -w2 y(t) a < 0 v > 0 a < 0 v < 0 t y(t) (A) (B) (C) a > 0 v > 0 The answer is (C).

Clicker Question A mass hanging from a vertical spring is lifted a distance d above equilibrium and released at t = 0. Which of the following describes its velocity and acceleration as a function of time? A) v(t) = -vmax sin(wt) a(t) = -amax cos(wt) B) v(t) = vmax sin(wt) a(t) = amax cos(wt) k y d C) v(t) = vmax cos(wt) a(t) = -amax cos(wt) t = 0 m (both vmax and amax are positive numbers)

Clicker Question Since we start with the maximum possible displacement at t = 0 we know that: y = d cos(wt) k y d t = 0 m

At t = 0, y = 0, moving down Use energy conservation to find A

Or similarly