Which way will a spinning baseball curve (relative to freefall)?

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Which way will a spinning baseball curve
(relative to freefall)?
Physics 1710Warm-up Quiz

0
A up
?
C left
v
D right
B down

1. up
2. down
3. left
4. right

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Physics 1710Chapter 15 SHO

• Bernoullis Effect
• P Po - ½ ? vrelative 2

?
v
P Po - ½ ? (v R?)2
C left
v
P Po - ½ ? (v - R?)2
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Physics 1710Chapter 15 SHO

• 1' Lecture
• Simple Harmonic Motion is sinusoidal.
• The period is the reciprocal of the frequency.
• For a mass m on a spring of spring constant k,
  the period T 2pv(m/k)
• For Damped SHO, the frequency is decreased and
  the amplitude decays exponentially.

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Physics 1710Chapter 15 SHO

• Existential Physics Questions
• What happens when you stretch a spring?
• What happens if you stretch the spring twice as
  much? Or three times as much?

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Physics 1710Chapter 15 SHO
F
x
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Physics 1710Chapter 15 SHO
x F 0 0
F
.1 m -1 N
.2 m -2 N
.3 m -3 N
x
F/x - 10 N/m
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Physics 1710Chapter 15 SHO

• Hookes Law
• When you stretch or compress a spring, the
  force (F) produced is proportional to the
  displacement (x) and in the direction to restore
  the system (-) to the original position.
• F - k x

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Physics 1710Chapter 15 SHO

• Spring Constant k
• The restoring force produced by a spring is
  proportional to the negative of its extension (or
  compression).
• F - k x (Hookes Law)
• F restoring force
• k spring constant
• x extension, the stretch or squeeze (if
  negative).

What is the restoring force of a spring with k
25. N/m when it is stretch by 10 cm?
F - (25. N/m )(0.10 m) -2.5 N
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Physics 1710Chapter 15 SHO
Mass on a Spring
Spring ?
Mass ?
Force
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Physics 1710Chapter 15 SHO
F
x
F/x - 10 N/m
F/x - 25. N/m -k
F/x - 12.5 N/m
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Physics 1710Chapter 15 SHO
What determines the frequency of the oscillation
of a simple harmonic oscillator?
x
The stronger the force (k), the more rapid is the
oscillation. The greater the mass (m) the slower
is the oscillation.
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Physics 1710Chapter 15 SHO

• Mass on a Spring (Quantitative)
• Fx - kx
• F ma m(d 2x/dt 2)
• So d 2x/dt 2 - (k/m) x (The Helmholtz
  equation)

Solution? Try x Xo cos(?t f) d 2(Xo cos(?t
f))/dt 2 - (k/m) Xo cos(?t f)? - Xo ?2
cos(?t f) - (k/m) Xo cos(?t f) iff ?2
k/m ? ? v (k/m) 2p f 2p/T
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Physics 1710Chapter 15 SHO

• Peregrination in Existential Physics

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Physics 1710Chapter 15 SHO

• Peregrination in Existential Physics

Cosine of the angle
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Physics 1710Chapter 15 SHO

• Mass on a Spring (Quantitative)
• d 2x/dt 2 - (k/m) x
• Implies that x Xo cos(?t f)
• and ? v (k/m) 2p f 2p/T
• or T 2p v (m/k)

What is the period T of a mass of 0.5 kg on a
spring with spring constant k 25. N/m?
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What is the period T of a mass of 0.5 kg on a
spring with spring constant k 25. N/m?
Physics 1710Chapter 15 SHO

1. 7.1 sec
2. 44. sec
3. 0.14 sec
4. 0.89 sec
5. None of the above

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Physics 1710Chapter 15 SHO
Lets Do Physics! Lets Do Physics! Lets Do
Physics!
Lets Do Physics! Yeah!

18
Physics 1710Chapter 14 Fluid Dynamics

• What will happen to the period if I double the
  mass?

Peer Instruction Time
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What will happen to the period if I double the
mass?
Physics 1710Chapter 15 SHO

1. It will stay the same.
2. It will increase by 2X
3. It will increase by v2X
4. It will decrease by ½X
5. It will decrease by 1/v2X

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Physics 1710Chapter 15 SHO
Lets Do Physics! Lets Do Physics! Lets Do
Physics!
Lets Do Physics! Yeah!

21
Physics 1710Chapter 15 SHO

• What is the velocity in Simple Harmonic Motion?
• x Xo cos(?t f)
• v dx/dt - ? Xo sin (?t f) - ? Xo cos
  (?t f p/2)
• N.B. In general in Xo cos(?tf)
• Xo is called the amplitude.
• ? v(k/m) 2p f 2p/T is the angular
  frequency.
• f is the phase.

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Physics 1710Chapter 15 SHO

• What is the Kinetic and Potential Energy ?
• x Xo cos(?t f)
• v dx/dt - ? Xo sin(?t f) - ? Xo cos(?t
  f p/2)
• K ½ mv2 ½ m ?2Xo2 sin2 (?t f) K ½ k Xo2
  sin2 (?t f) , since ?2 k/m
• U ½ k x2 ½ k Xo2 sin2(?t f)
• Total E KU ½ k Xo2 cos2(?t f) sin2(?t
  f)
• E ½ k Xo2

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Physics 1710Chapter 15 SHO

• Damping

What if there is a viscous drag Fdamping -b
v? Then md 2x/dt 2 - k x b dx/dt Try x
Xo e ½ (b/m)t cos(?t f) dx/dt ( ½ b/m) Xo
e ½ (b/m)t cos(?t f) - ?Xo e ½ b/m
sin(?t f) d 2x/dt 2 (½ b/m)2 Xo e ½ (b/m)t
cos(?t f) 2( ½ b/m) (- ?)Xo e ½ (b/m)t
sin(?t f) - ?2 Xo e ½ (b/m)t cos(?t f)
(½ b/m)2 - ?2 x b/m (dx/dt), okay for ?
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Physics 1710Chapter 15 SHO

• Damping

x Xo e ½ (b/m)t cos(?t f) for ? vk/m (½
b/m)2
Energy E ½ kXo2 e (b/m)t
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Physics 1710Chapter 15 SHO

• Summary
• Simple Harmonic Motion is sinusoidal. x
  Xo cos(?t f)
• The period is the reciprocal of the
  frequency. T 1/ f
• For a mass m on a spring of spring constant k,
  the period T 2pv(m/k)
• For Damped SHO, the frequency is decreased and
  the amplitude decays exponentially.
• x Xo e ½ (b/m)t cos(?t f)with ? vk/m ½
  b/m