Lecture 39 1

2 What is the sign of cos(225 o )? Sign of sin(225 o )? Don’t use a calculator! A) cos(225 o ) = (+), sin(225 o ) = (–) B) cos(225 o ) = (–), sin(225 o ) = (+) C) cos(225 o ) = (+), sin(225 o ) = (+) D) cos(225 o ) = (–), sin(225 o ) = (–) E) None of these. One of them is zero Clicker Question Room Frequency BA

CAPA assignment #14 is due on Friday at 10 pm. This week in Section: Lab #6 (with prelab) Read Chapter 12 on Sound Announcements 3

4 Horizontal Spring and Mass Oscillation Amplitude A = E x = A, v = 0x = 0, v = ±v max At turning points x = ±A, v = 0, E T = only PE= At equilibrium point x = 0, v =±v max, E T = only KE= x = -A, v = 0

5 Harmonic Time Dependence of SHM A +A –A 0 vmvm  x SHM is mathematically the same as one component of circular motion at constant speed v m, with ω is constant and θ = ωt. x = A cos ωt at t = 0 x = A sin ωt For horizontal mass m oscillating with spring, spring constant k

6 270 o = (3/2)π. What is cos[(3/2) π] and what is sin[(3/2) π] ? A) cos[(3/2) π] = 1, sin[(3/2) π] = 0 B) cos[(3/2) π] = 0, sin[(3/2) π] = 1 C) cos[(3/2) π] = 1, sin[(3/2) π] = 1 D) cos[(3/2) π] = 0, sin[(3/2) π] = 0 E) None of these Clicker Question Room Frequency BA cos[(3/2) π] = 0, sin[(3/2) π] = -1

7 Simple Harmonic Oscillator with x = 0 at t = 0

8 The position of a mass on a spring as a function of time is shown below. When the mass is at point P on the graph A) The velocity v > 0 B) v < 0 C) v = 0 Clicker Question Room Frequency BA v is the slope at P

9 The position of a mass on a spring as a function of time is shown below. When the mass is at point P on the graph A) The acceleration a > 0 B) a < 0 C) a = 0 Clicker Question Room Frequency BA As the mass approaches its extreme position, it is slowing down (velocity positive but decreasing) so the acceleration must be negative

10 Vertical Spring and Mass Oscillation Spring Force Gravity y Spring equilibrium without gravity y=0 Spring equilibrium with gravity y = y E New spring equilibrium length where –mg - ky E = 0 y E = -mg/k Oscillation frequency is NOT changed! ω 2 = k/m

11 Vertical Spring and Mass Oscillation y Spring equilibrium without gravity y=0 Spring equilibrium with gravity y = y E Energy Still Conserved! Now E T has gravity PE term: With a little algebra you can rewrite this as

12 Simple Pendulum Oscillation mg There is a net force back towards the vertical equilibrium position! This gives oscillation, but is it SHM? θ s =θL L m θ T = mg cosθ F net = -mg sin θ No! For pure SHM we would need F net = -mg θ BUT! For small θ, sin θ ≈ θ = s/L, so we get ma tan = F net ≈ -mgs/L

13 Simple Pendulum Oscillation for Small Angles We found ma tan ≈ -mgs/L for small θ Cancelling m gives a tan ≈ -(g/L)s For horizontal spring we had a x = -(k/m)x Use SHM formulas with g/L in place of k/m !!! Frequency independent of amplitude θ 0 Frequency independent of mass m

14 Will a given pendulum have a shorter or longer or equal period on the moon compared to the period on earth? A) Equal periods B) Shorter on MoonC) Longer on Moon Clicker Question Room Frequency BA “g” is smaller on the moon so T is longer

15 The Physical Pendulum Any object suspended from any point in the object except the center-of-mass will swing back and forth! This is called a physical pendulum, as opposed to a simple pendulum. “L” is now distance from pivot to Center of Mass Now, changing distribution of mass will change period, frequency, if the Center of Mass is changed

16 Waves are Everywhere!!! Whenever you have a bunch of stuff or many things which can interact with each other, you can get waves.

17 Wave Simulation A great simulation to learn about one-dimensional waves can be found at