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Lecture 39

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

Clicker Question Room Frequency BA

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• CAPA assignment #14 is due on Friday at 10 pm.

• This week in Section: Lab #6 (with prelab)

• Read Chapter 12 on Sound

Announcements

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Horizontal Spring and Mass Oscillation

ET =12 kx

2 + 12 mv

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Amplitude A = E

x = A, v = 0x = 0, v = ±vmax

At turning points x = ±A, v = 0, ET = only PE= 12 kA

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At equilibrium point x = 0, v =±vmax, ET = only KE= 12 mvmax2 .

x = -A, v = 0

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Harmonic Time Dependence of SHM

A

+A–A

0

vm

θ

x

SHM is mathematically the same as one component of circular motion at constant speed vm, with ω is constant and θ = ωt.

x = A cos ωt at t = 0

at t = 0x = A sin ωt

ω =k

m (radians/sec)

For horizontal mass m oscillating with spring, spring constant k

f =12π

km

T =2π mk

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270o = (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] = 0B) cos[(3/2) π] = 0, sin[(3/2) π] = 1C) cos[(3/2) π] = 1, sin[(3/2) π] = 1D) cos[(3/2) π] = 0, sin[(3/2) π] = 0E) None of these

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cos[(3/2) π] = 0, sin[(3/2) π] = -1

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Simple Harmonic Oscillator with x = 0 at t = 0

x =Asinωt = Asin(2π ft) = Asin(2πt /T )

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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

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v is the slope at P

x

t

P

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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

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As the mass approaches its extreme position, it is slowing down (velocity positive but decreasing) so the acceleration must be negative x

t

P

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Vertical Spring and Mass Oscillation

Spring Force

Gravity

y

Spring equilibrium without gravity y=0

Spring equilibrium with gravity y = yE

New spring equilibrium length where –mg - kyE = 0

yE = -mg/k

Oscillation frequency is NOT changed! ω2 = k/m

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Vertical Spring and Mass Oscillation

y

Spring equilibrium without gravity y=0

Spring equilibrium with gravity y = yE

Energy Still Conserved! Now ET has gravity PE term:

ET =12 ky

2 + 12 mv

2 +mgyWith a little algebra you can rewrite this as

ET =12 k(y−yE )

2 + 12 mv

2 + 12 kyE

2

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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

Fnet = -mg sin θ

No! For pure SHM we would need Fnet = -mg θ

BUT! For small θ, sin θ ≈ θ = s/L, so we get matan = Fnet ≈ -mgs/L

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Simple Pendulum Oscillation for Small Angles

We found matan ≈ -mgs/L for small θ

Cancelling m gives atan ≈ -(g/L)sFor horizontal spring we had ax = -(k/m)x

Use SHM formulas with g/L in place of k/m !!!

ω =g

L (radians/sec) f =

12π

gL

T =2π Lg

θ =θ0 cosωt or θ =θ 0 sinωt

• Frequency independent of amplitude θ0

• Frequency independent of mass m

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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 Moon C) Longer on Moon

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“g” is smaller on the moon so T is longerT =2π Lg

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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

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Waves are Everywhere!!!

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

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Wave Simulation

A great simulation to learn about one-dimensional waves can be found at http://phet.colorado.edu/en/simulation/wave-on-a-string