physics 1022: chapter 14 waves - the george washington...

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Phys 1022: Introduction, Pg 1


Physics 1022: Chapter 14

Waves

  Anatomy of a wave

  Simple harmonic motion

  Energy and simple harmonic motion

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Phys 1022: Introduction, Pg 3 New Topic

Waves


Simple Harmonic Motion: The restoring force is proportional to the negative of the displacement (like F=-kx)

Fma =

kxdtxdm −=2

2

02

2

=+ xmk

dtxd

A is amplitude, φ is phase angle. They are determined by initial conditions (the value of x and v at t=0.)

)cos( φω += tAxgeneral solution:

mkx

dtxd

==+ ωω with ,022

2

3


Phase angle

A measure of different starting positions (and velocities)

)cos( φω += tAx

φω

φ

sin,cos

:0tAt

0

0

AvAx−=

=

=

)sin( φωω +−== tAdtdxv

Both A and φ can be determined by initial conditions:

( )

ωφ

ω

0

0

20

20

tan

,/

xvvxA

−=

+=


The graph shows a particle in SHM.

(a) What is the phase constant φ0?

(b) What is the phase of the particle at the numbered points on the graph?

(c) Place dots (with labels) on the circle to indicate the particle’s position corresponding to the numbered points.

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A particle moves CCW around a circle at constant speed. From the phase constants, show the particle’s initial position and sketch two cycles of the x vs. t graph.

What does the v vs. t graph look like?


Simple Harmonic ↔ Circular Motion If we look at uniform circular motion from the side, the object appears to

move in simple harmonic motion

A

Top View:

x

y Consider the x-position of the object:

v = v0 vx = 0

v = v0 vx = v0

x θ

x = Acos θ now θ = ω t

x = Acos (ωt)

Side View:

x = +A

y Click here for demo

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ConcepTest 4(Post)Energy in SHM

This is the potential energy diagram of a particle oscillating on a spring. What is the equilibrium length of the spring?

1.  12 cm

2.  16 cm

3.  20 cm

4.  24 cm 5.  28 cm

6.  cannot be determined from the graph


ConcepTest 5(Post)Energy in SHM If the particle’s turning points are 14 cm and 26 cm, draw a line that indicates the total energy and then determine the particle’s maximum kinetic energy.

Etot ~ 6.5 J

1.  2.5 J

2.  5 J

3.  7.5 J

4.  10 J 5.  12.5 J

6.  15 J 7.  other

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Energy in Simple Harmonic Motion

m

equilibrium position

m

x k

Frestore

  Energy of the oscillating system at any time is constant: Etotal = PEspring + KEmass

m

xmax = A k

Frestore

  at end, x = A and v = 0 (KE = 0) Etotal = PEspring = 1/2kA2

  at any point in between … Etotal = 1/2kx2 + 1/2mv2

  at x = 0 , PE = 0 and v = v0 Etotal = KEmass = 1/2mv0

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ENERGY IS CONSERVED


Draw a graph of the particle’s kinetic energy as a function of position.

What will be the turning points if the particle’s total energy is doubled?

2max2

1221 mvkA =

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ConcepTest 6(Post)Simple Harmonic Motion A block oscillating on a spring has a period of T = 4 s. If the mass of the block is halved, what is the new period?

1.  1 s

2.  2 s

3.  2.8 s

4.  4 s 5.  5.6 s

6.  8 s 7.  16 s


ConcepTest 7(Post)Simple Harmonic Motion

A block oscillating on a spring has a period of T = 4 s. If the spring constant is quadrupled, what is the new period?

1.  1 s

2.  2 s

3.  2.8 s

4.  4 s 5.  5.6 s

6.  8 s 7.  16 s

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ConcepTest 8(Post)Simple Harmonic Motion

A block oscillating on a spring has a period of T = 4 s. If the oscillation amplitude is doubled, what is the new period?

1.  1 s

2.  2 s

3.  2.8 s

4.  4 s 5.  5.6 s

6.  8 s 7.  16 s


(a) Draw the v vs. t and the a vs. t graphs.

(b) When x is greater than zero, is a ever greater than zero? When?

(c) When x is greater than zero, is v ever greater than zero? When?

Can we describe the entire motion of this oscillating system?

The graph shows x vs. t for a particle in SHM.

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Based on this “reference” graph of x vs. t, draw the new graphs that represent the following conditions:

(a) amplitude and frequency are doubled

(b) amplitude is halved and mass is quadrupled

(c) phase constant is increased by π/2 rad

(d) max. speed is doubled while amplitude stays constant


Simple Pendulum

New Topic

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ConcepTest 9(Post)Simple Pendulum A pendulum on Planet X, where the value of g is unknown, oscillates with a period of T = 4 s. If the mass is quadrupled, what is the new period?

1.  1 s

2.  2 s

3.  2.8 s

4.  4 s 5.  5.6 s

6.  8 s 7.  16 s


The Simple Pendulum

Consider only small oscillations ⇒ sinθ ≈ θ

F = –mg sin θ = - mg θ = –mg x/L

= –(mg/L) x

Period of a pendulum does NOT depend on ý  mass

ý  amplitude

(Try this on your calculator, but θ must be in radians!)

T = 2π √ m / k

F = –kx

Spring Oscillator

F = –mg/L x

T = 2π √ m / (mg/L) T = 2π √ L / g

Pendulum

Restoring force:

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ConcepTest 12(Post)Simple Pendulum

If a pendulum with period T on Earth is taken to the Moon, how will the period change?

1.  increases

2.  decreases

3.  stays the same

4.  no way to tell



If a mass-spring system with period T on Earth is taken to the Moon, how will the period change?

1.  increases

2.  decreases



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A pendulum in an elevator has period T when the elevator is at rest. If the elevator is accelerating upward, how will the period change?

What happens to the period of the pendulum if the elevator is in free fall?

1.  increases

2.  decreases




ConcepTest 15(Post)To the Center of the Earth A hole is drilled through the center of the Earth and emerges on the other side. You jump into the hole. What happens to you?

1.  you fall to the center and stop

2.  you go all the way through and continue off into space

3.  you fall to the other side of the Earth and stay there

4.  you fall to the other side of the Earth and then return

5.  you won’t fall at all

physics 1022: chapter 14 waves - the george washington...

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