simple harmonic motion

Simple Harmonic MotionSimple Harmonic Motion

Simple Harmonic MotionSimple Harmonic Motion Vibrations

Vocal cords when singing/speakingString/rubber band

Simple Harmonic MotionRestoring force proportional to displacementSprings F = -kx

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Question IIQuestion IIA mass on a spring oscillates back & forth with simple harmonic motion of amplitude A. A plot of displacement (x) versus time (t) is shown below. At what points during its oscillation is the magnitude of the acceleration of the block biggest?

1. When x = +A or -A (i.e. maximum displacement)

2. When x = 0 (i.e. zero displacement)

3. The acceleration of the mass is constant

+A

t-A

x

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Potential Energy in SpringPotential Energy in Spring Force of spring is Conservative

F = -k xW = -1/2 k x2

Work done only depends on initial and final position

Define Potential Energy PEspring = ½ k x2

Force

x

work

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***Energy in SHM******Energy in SHM*** A mass is attached to a spring and set to

motion. The maximum displacement is x=AEnergy = PE + KE = constant!

= ½ k x2 + ½ m v2

At maximum displacement x=A, v = 0Energy = ½ k A2 + 0

At zero displacement x = 0

Energy = 0 + ½ mvm2

Since Total Energy is same

½ k A2 = ½ m vm2

vm = sqrt(k/m) Am

xx=0

0x

PES

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Question 3Question 3A mass on a spring oscillates back & forth with simple harmonic motion of amplitude A. A plot of displacement (x) versus time (t) is shown below. At what points during its oscillation is the speed of the block biggest?

1. When x = +A or -A (i.e. maximum displacement)

2. When x = 0 (i.e. zero displacement)

3. The speed of the mass is constant

+A

t-A

x

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Question 4Question 4 A spring oscillates back and forth on a frictionless horizontal

surface. A camera takes pictures of the position every 1/10th of a second. Which plot best shows the positions of the mass.

EndPoint EndPointEquilibrium

EndPoint EndPointEquilibrium

EndPoint EndPointEquilibrium38

1

2

3

X=0

X=AX=-A

X=A; v=0; a=-amax

X=0; v=-vmax; a=0

X=-A; v=0; a=amax

X=0; v=vmax; a=0

X=A; v=0; a=-amax

Springs and Simple Harmonic Springs and Simple Harmonic MotionMotion

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What does moving in a circle have to do with moving back & forth in a straight line ??

y

x

-R

R

0

1 1

2 2

3 3

4 4

5 5

6 62

R

8

7

8

7

23

x

x = R cos = R cos (t)since = t

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SHM and CirclesSHM and Circles

Simple Harmonic Motion:Simple Harmonic Motion:

x(t) = [A]cos(t)

v(t) = -[A]sin(t)

a(t) = -[A2]cos(t)

x(t) = [A]sin(t)

v(t) = [A]cos(t)

a(t) = -[A2]sin(t)

xmax = A

vmax = A

amax = A2

Period = T (seconds per cycle)

Frequency = f = 1/T (cycles per second)

Angular frequency = = 2f = 2/T

For spring: 2 = k/m

OR

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ExampleExample

A 3 kg mass is attached to a spring (k=24 N/m). It is stretched 5 cm. At time t=0 it is released and oscillates.

Which equation describes the position as a function of time x(t) =

A) 5 sin(t) B) 5 cos(t) C) 24 sin(t)

D) 24 cos(t) E) -24 cos(t)

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ExampleExample


What is the total energy of the block spring system?

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ExampleExample


What is the maximum speed of the block?

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ExampleExample


How long does it take for the block to return to x=+5cm?

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Pendulum MotionPendulum Motion For small angles

T = mgTx = -mg (x/L) Note: F proportional to x!

Fx = m ax

-mg (x/L) = m ax

ax = -(g/L) xRecall for SHO a = -2 x

= sqrt(g/L) T = 2 sqrt(L/g)

Period does not depend on A, or m!

m

L

x

T

mg

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Example: ClockExample: Clock If we want to make a grandfather clock so that the

pendulum makes one complete cycle each sec, how long should the pendulum be?

Question 1Question 1Suppose a grandfather clock (a simple pendulum) runs slow. In order to make it run on time you should:

1. Make the pendulum shorter

2. Make the pendulum longer

Lg

gL

T 2

2

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SummarySummary Simple Harmonic Motion

Occurs when have linear restoring force F= -kx x(t) = [A] cos(t)v(t) = -[A] sin(t)a(t) = -[A] cos(t)

SpringsF = -kxU = ½ k x2

= sqrt(k/m) Pendulum (Small oscillations)

= sqrt(L/g)50

simple harmonic motion

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