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Example of a SHO?: Pendulum

Ignoring friction, is the pendulum a SHO? (Recall that a SHO has the characteristic that the restoring force is proportional to displacement.) A)  Yes B)  No C)  Impossible to tell from

information given.

Approximately, for small amplitude oscillations.

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Assignments

Announcements: •  CAPA 14 is due Friday at midnight. It contains 6 problems

on Simple Harmonic Motion (CH. 13) and 11 questions on review.

•  HW13 is due during recitation this week.

Today: •  Finish up Simple Harmonic Motion: Ch. 13. •  Move on to a review of the semester via clicker questions

today and Friday.

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Assignments

The Final Exam: •  Monday Dec 15, 4:30 – 7:00 PM. Family Names A-R: Coors Event Center, Sections 15-23. Family Names S-Z: MATH 100. •  The final exam will cover material only up through Ch. 13. •  Material on D2L available for you to prepare: Last semester’s final exam. Prof Dubson’s final exam review video. Lecture slides and videos (on course web site). Prof Dubson’s chapter notes. CAPA solutions. This semester’s and past semester’s midterm exams.

Final Exam  

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Location & Time of the Final Exam: Mon Dec 15, 4:30 – 7:00 PM

Family Names A-R: Coors Event Center When you arrive, pick up a lap board, find a seat that has an exam booklet & bubble sheet, in sections 15-23.

Family Name S-Z: MATH 100. Special Accommodations: Students should have received an email from Prof Munsat with time and location. Prof Radzihovsky will proctor.

Sections 15-23

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Simple Harmonic Oscillator & Conservation of Energy

0

+x

-x k

m

Neglecting friction (no damping) and assuming there are no external forces (no driving):

Energy is Conserved

Etot = KE + PE = 12mv2 + 1

2kx2

6  

Simple Harmonic Oscillator & Conservation of Energy

PE = 12kx2

KEmax =12mvmax

2

PEmax =12kA2

x = 0 PE = 0 KE = 12mvmax

2 = Etot

x = ±A KE = 0 PE = 12kA2 = Etot

⎫

⎬⎪⎪

⎭⎪⎪

⇒ vmax =kmA =ωA

Already saw this

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7  

Simple Harmonic Oscillator & Conservation of Energy

x(t) = Acos(ωt +ϕ ) xmax = A

v(t) = −Aω sin(ωt +ϕ ) vmax =ωA = kmA

a(t) = −Aω 2 cos(ωt +ϕ ) amax =ω2A = k

mA

Showed earlier:

L43  W  12/10/14  a+er  lecture  

Displacement = x ≈ L sinθ

Restoring force = −mgsinθ = − mgL

⎛⎝⎜

⎞⎠⎟ x = −keff x

keff = Effective spring constant = mgL

ω =keffm

= mgLm

= gL

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Example of a SHO: Pendulum

θ

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ω =gL

f = ω2π

=1

2πgL

T =1f= 2π L

g

Example of a SHO: Pendulum

-- note: it’s independent of mass

A pendulum is only approximately a SHO. Its period actually depends weakly on its amplitude: Pendulum Lab (PhET)

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ω =gL

f = ω2π

=1

2πgL

T =1f= 2π L

g

rad/s

Hz = 1/sec

sec

Example of a SHO: Pendulum

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A child is swinging on a swing with a period T. A second child climbs on with the first, doubling the weight on the swing. The period of the swing is now... A)  the same, T B)  2T C)  D)  2π T.

T = 2π Lg

2 TThe swing’s period is independent of mass.

Begin  Review  

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Q

−R

y

x

y

Q −R

Tip-to-Tail Vector Addition

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W  

v

Fg

Wup < 0

Fg v

Wdown > 0

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v

Fdrag

Wup < 0

v

Wdown < 0

Fdrag

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(heavy) (light)