April 29, Week 15 Today: Chapter 14, Periodic Motion Homework - - PowerPoint PPT Presentation

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April 29, Week 15 Today: Chapter 14, Periodic Motion Homework - - PowerPoint PPT Presentation

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April 29, Week 15 Today: Chapter 14, Periodic Motion Homework Assignment #11 - Due May 3. Mastering Physics: 7 questions from chapters 13 and 14. Mastering Physics: 13.77 Exams will be graded by Wednesday. Newtons Gravity April 29, 2013 - p.

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Newton’s Gravity April 29, 2013 - p. 1/12

April 29, Week 15

Today: Chapter 14, Periodic Motion Homework Assignment #11 - Due May 3.

Mastering Physics: 7 questions from chapters 13 and 14. Mastering Physics: 13.77

Exams will be graded by Wednesday.

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Newton’s Gravity April 29, 2013 - p. 2/12

Periodic Motion

Periodic Motion or Oscillation - Any repeated motion.

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Newton’s Gravity April 29, 2013 - p. 2/12

Periodic Motion

Periodic Motion or Oscillation - Any repeated motion. Terms:

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Newton’s Gravity April 29, 2013 - p. 2/12

Periodic Motion

Periodic Motion or Oscillation - Any repeated motion. Terms: Cycle - One complete round trip.

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SLIDE 5

Newton’s Gravity April 29, 2013 - p. 2/12

Periodic Motion

Periodic Motion or Oscillation - Any repeated motion. Terms: Cycle - One complete round trip. Amplitude, A - Maximum displacement from zero.

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Newton’s Gravity April 29, 2013 - p. 2/12

Periodic Motion

Periodic Motion or Oscillation - Any repeated motion. Terms: Cycle - One complete round trip. Amplitude, A - Maximum displacement from zero. Period, T - Time for one cycle.

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SLIDE 7

Newton’s Gravity April 29, 2013 - p. 2/12

Periodic Motion

Periodic Motion or Oscillation - Any repeated motion. Terms: Cycle - One complete round trip. Amplitude, A - Maximum displacement from zero. Period, T - Time for one cycle. Frequency, f - The number of cycles per time.

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SLIDE 8

Newton’s Gravity April 29, 2013 - p. 2/12

Periodic Motion

Periodic Motion or Oscillation - Any repeated motion. Terms: Cycle - One complete round trip. Amplitude, A - Maximum displacement from zero. Period, T - Time for one cycle. Frequency, f - The number of cycles per time. f = 1 T

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SLIDE 9

Newton’s Gravity April 29, 2013 - p. 2/12

Periodic Motion

Periodic Motion or Oscillation - Any repeated motion. Terms: Cycle - One complete round trip. Amplitude, A - Maximum displacement from zero. Period, T - Time for one cycle. Frequency, f - The number of cycles per time. f = 1 T Unit: 1 s = Hz (Hertz)

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Newton’s Gravity April 29, 2013 - p. 3/12

Simple Harmonic Motion

Simple Harmonic Motion (SHM) - The simplest type of periodic

  • motion. Occurs when a mass is connected to a spring with no

friction.

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SLIDE 11

Newton’s Gravity April 29, 2013 - p. 3/12

Simple Harmonic Motion

Simple Harmonic Motion (SHM) - The simplest type of periodic

  • motion. Occurs when a mass is connected to a spring with no

friction.

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SLIDE 12

Newton’s Gravity April 29, 2013 - p. 3/12

Simple Harmonic Motion

Simple Harmonic Motion (SHM) - The simplest type of periodic

  • motion. Occurs when a mass is connected to a spring with no

friction.

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SLIDE 13

Newton’s Gravity April 29, 2013 - p. 3/12

Simple Harmonic Motion

Simple Harmonic Motion (SHM) - The simplest type of periodic

  • motion. Occurs when a mass is connected to a spring with no

friction. x

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SLIDE 14

Newton’s Gravity April 29, 2013 - p. 3/12

Simple Harmonic Motion

Simple Harmonic Motion (SHM) - The simplest type of periodic

  • motion. Occurs when a mass is connected to a spring with no

friction. − → F

el

x

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SLIDE 15

Newton’s Gravity April 29, 2013 - p. 3/12

Simple Harmonic Motion

Simple Harmonic Motion (SHM) - The simplest type of periodic

  • motion. Occurs when a mass is connected to a spring with no

friction. − → F

el

x − → F = m− → a

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SLIDE 16

Newton’s Gravity April 29, 2013 - p. 3/12

Simple Harmonic Motion

Simple Harmonic Motion (SHM) - The simplest type of periodic

  • motion. Occurs when a mass is connected to a spring with no

friction. − → F

el

x − → F = m− → a −Fel = max

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SLIDE 17

Newton’s Gravity April 29, 2013 - p. 3/12

Simple Harmonic Motion

Simple Harmonic Motion (SHM) - The simplest type of periodic

  • motion. Occurs when a mass is connected to a spring with no

friction. − → F

el

Fel = kx x − → F = m− → a −Fel = max

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SLIDE 18

Newton’s Gravity April 29, 2013 - p. 3/12

Simple Harmonic Motion

Simple Harmonic Motion (SHM) - The simplest type of periodic

  • motion. Occurs when a mass is connected to a spring with no

friction. − → F

el

Fel = kx x − → F = m− → a −Fel = max −kx = max

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SLIDE 19

Newton’s Gravity April 29, 2013 - p. 3/12

Simple Harmonic Motion

Simple Harmonic Motion (SHM) - The simplest type of periodic

  • motion. Occurs when a mass is connected to a spring with no

friction. − → F

el

Fel = kx x − → F = m− → a −Fel = max −kx = max ax = − k mx

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SLIDE 20

Newton’s Gravity April 29, 2013 - p. 4/12

Simple Harmonic Motion II

Simple Harmonic Motion (SHM) - The simplest type of periodic

  • motion. Occurs when a mass is connected to a spring with no

friction. − → F

el

Fel = kx x −kx = max

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SLIDE 21

Newton’s Gravity April 29, 2013 - p. 4/12

Simple Harmonic Motion II

Simple Harmonic Motion (SHM) - The simplest type of periodic

  • motion. Occurs when a mass is connected to a spring with no

friction. − → F

el

Fel = kx x −kx = max vx = dx dt

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SLIDE 22

Newton’s Gravity April 29, 2013 - p. 4/12

Simple Harmonic Motion II

Simple Harmonic Motion (SHM) - The simplest type of periodic

  • motion. Occurs when a mass is connected to a spring with no

friction. − → F

el

Fel = kx x −kx = max ax = dvx dt vx = dx dt

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SLIDE 23

Newton’s Gravity April 29, 2013 - p. 4/12

Simple Harmonic Motion II

Simple Harmonic Motion (SHM) - The simplest type of periodic

  • motion. Occurs when a mass is connected to a spring with no

friction. − → F

el

Fel = kx x −kx = max ax = d2x dt2 ax = dvx dt vx = dx dt

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Newton’s Gravity April 29, 2013 - p. 5/12

Simple Harmonic Motion III

Simple Harmonic Motion (SHM) - The simplest type of periodic

  • motion. Occurs when a mass is connected to a spring with no

friction. − → F

el

Fel = kx x −kx = md2x dt2 ax = d2x dt2 ax = dvx dt vx = dx dt

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Newton’s Gravity April 29, 2013 - p. 5/12

Simple Harmonic Motion III

Simple Harmonic Motion (SHM) - The simplest type of periodic

  • motion. Occurs when a mass is connected to a spring with no

friction. − → F

el

Fel = kx x −kx = md2x dt2 d2x dt2 = − k m

  • x

Differential Equation for SHM

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SLIDE 26

Newton’s Gravity April 29, 2013 - p. 6/12

Simple Harmonic Motion IV

Simple Harmonic Motion (SHM) - The simplest type of periodic

  • motion. Occurs when a mass is connected to a spring with no

friction. − → F

el

Fel = kx x d2x dt2 = − k m

  • x
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SLIDE 27

Newton’s Gravity April 29, 2013 - p. 6/12

Simple Harmonic Motion IV

Simple Harmonic Motion (SHM) - The simplest type of periodic

  • motion. Occurs when a mass is connected to a spring with no

friction. − → F

el

Fel = kx x d2x dt2 = − k m

  • x

In Calculus: f ′′ = −cf

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SLIDE 28

Newton’s Gravity April 29, 2013 - p. 6/12

Simple Harmonic Motion IV

Simple Harmonic Motion (SHM) - The simplest type of periodic

  • motion. Occurs when a mass is connected to a spring with no

friction. − → F

el

Fel = kx x d2x dt2 = − k m

  • x

In Calculus: f ′′ = −cf x = A cos (ωt + φ)

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SLIDE 29

Newton’s Gravity April 29, 2013 - p. 6/12

Simple Harmonic Motion IV

Simple Harmonic Motion (SHM) - The simplest type of periodic

  • motion. Occurs when a mass is connected to a spring with no

friction. − → F

el

Fel = kx x d2x dt2 = − k m

  • x

In Calculus: f ′′ = −cf x = A cos (ωt + φ) Amplitude

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SLIDE 30

Newton’s Gravity April 29, 2013 - p. 6/12

Simple Harmonic Motion IV

Simple Harmonic Motion (SHM) - The simplest type of periodic

  • motion. Occurs when a mass is connected to a spring with no

friction. − → F

el

Fel = kx x d2x dt2 = − k m

  • x

In Calculus: f ′′ = −cf x = A cos (ωt + φ) Amplitude

Phase Angle

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SLIDE 31

Newton’s Gravity April 29, 2013 - p. 6/12

Simple Harmonic Motion IV

Simple Harmonic Motion (SHM) - The simplest type of periodic

  • motion. Occurs when a mass is connected to a spring with no

friction. − → F

el

Fel = kx x d2x dt2 = − k m

  • x

In Calculus: f ′′ = −cf x = A cos (ωt + φ) Amplitude Angular frequency, ω = 2πf = 2π T

Phase Angle

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Newton’s Gravity April 29, 2013 - p. 7/12

Amplitude

Amplitude - Maximum distance from zero.

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Newton’s Gravity April 29, 2013 - p. 7/12

Amplitude

Amplitude - Maximum distance from zero. x = cos t

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Newton’s Gravity April 29, 2013 - p. 7/12

Amplitude

Amplitude - Maximum distance from zero. x = cos t

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Newton’s Gravity April 29, 2013 - p. 7/12

Amplitude

Amplitude - Maximum distance from zero. x = cos t

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Newton’s Gravity April 29, 2013 - p. 7/12

Amplitude

Amplitude - Maximum distance from zero. x = cos t 1 −1

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Newton’s Gravity April 29, 2013 - p. 7/12

Amplitude

Amplitude - Maximum distance from zero. x = cos t 1 −1 x = A cos t

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Newton’s Gravity April 29, 2013 - p. 7/12

Amplitude

Amplitude - Maximum distance from zero. x = cos t 1 −1 x = A cos t

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Newton’s Gravity April 29, 2013 - p. 7/12

Amplitude

Amplitude - Maximum distance from zero. x = cos t 1 −1 x = A cos t

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Newton’s Gravity April 29, 2013 - p. 7/12

Amplitude

Amplitude - Maximum distance from zero. x = cos t 1 −1 x = A cos t A −A

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Newton’s Gravity April 29, 2013 - p. 8/12

Phase Angle

Phase Angle - φ, Units: rad. Shifts the Cosine to start wherever needed.

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Newton’s Gravity April 29, 2013 - p. 8/12

Phase Angle

Phase Angle - φ, Units: rad. Shifts the Cosine to start wherever needed. x = A cos t

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Newton’s Gravity April 29, 2013 - p. 8/12

Phase Angle

Phase Angle - φ, Units: rad. Shifts the Cosine to start wherever needed. x = A cos t x = A cos

  • t + π

4

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Newton’s Gravity April 29, 2013 - p. 8/12

Phase Angle

Phase Angle - φ, Units: rad. Shifts the Cosine to start wherever needed. x = A cos t x = A cos

  • t + π

4

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Newton’s Gravity April 29, 2013 - p. 8/12

Phase Angle

Phase Angle - φ, Units: rad. Shifts the Cosine to start wherever needed. x = A cos t x = A cos

  • t + π

4

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Newton’s Gravity April 29, 2013 - p. 8/12

Phase Angle

Phase Angle - φ, Units: rad. Shifts the Cosine to start wherever needed. x = A cos t x = A cos

  • t + π

4

  • x = A cos
  • t + π

2

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Newton’s Gravity April 29, 2013 - p. 8/12

Phase Angle

Phase Angle - φ, Units: rad. Shifts the Cosine to start wherever needed. x = A cos t x = A cos

  • t + π

4

  • x = A cos
  • t + π

2

  • = − sin t
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Newton’s Gravity April 29, 2013 - p. 9/12

Angular Frequency

Angular Frequency - ω = 2πf = 2π T Units: rad/s

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Newton’s Gravity April 29, 2013 - p. 9/12

Angular Frequency

Angular Frequency - ω = 2πf = 2π T Units: rad/s x = A cos t

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Newton’s Gravity April 29, 2013 - p. 9/12

Angular Frequency

Angular Frequency - ω = 2πf = 2π T Units: rad/s x = A cos t

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Newton’s Gravity April 29, 2013 - p. 9/12

Angular Frequency

Angular Frequency - ω = 2πf = 2π T Units: rad/s x = A cos t

T = 2π

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Newton’s Gravity April 29, 2013 - p. 9/12

Angular Frequency

Angular Frequency - ω = 2πf = 2π T Units: rad/s x = A cos t

T = 2π

x = A cos (2πt)

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Newton’s Gravity April 29, 2013 - p. 9/12

Angular Frequency

Angular Frequency - ω = 2πf = 2π T Units: rad/s x = A cos t

T = 2π

x = A cos (2πt)

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Newton’s Gravity April 29, 2013 - p. 9/12

Angular Frequency

Angular Frequency - ω = 2πf = 2π T Units: rad/s x = A cos t

T = 2π

x = A cos (2πt)

T = 1

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Newton’s Gravity April 29, 2013 - p. 9/12

Angular Frequency

Angular Frequency - ω = 2πf = 2π T Units: rad/s x = A cos t

T = 2π

x = A cos (2πt)

T = 1

x = A cos 2π

3 t

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SLIDE 56

Newton’s Gravity April 29, 2013 - p. 9/12

Angular Frequency

Angular Frequency - ω = 2πf = 2π T Units: rad/s x = A cos t

T = 2π

x = A cos (2πt)

T = 1

x = A cos 2π

3 t

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SLIDE 57

Newton’s Gravity April 29, 2013 - p. 9/12

Angular Frequency

Angular Frequency - ω = 2πf = 2π T Units: rad/s x = A cos t

T = 2π

x = A cos (2πt)

T = 1

x = A cos 2π

3 t

  • T = 3
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Newton’s Gravity April 29, 2013 - p. 10/12

General Solution

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Newton’s Gravity April 29, 2013 - p. 10/12

General Solution

x = A cos (ωt + φ)

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SLIDE 60

Newton’s Gravity April 29, 2013 - p. 10/12

General Solution

x = A cos (ωt + φ)

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Newton’s Gravity April 29, 2013 - p. 10/12

General Solution

x = A cos (ωt + φ) A −A

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SLIDE 62

Newton’s Gravity April 29, 2013 - p. 10/12

General Solution

x = A cos (ωt + φ) A −A

T = 2π ω

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SLIDE 63

Newton’s Gravity April 29, 2013 - p. 10/12

General Solution

x = A cos (ωt + φ) A −A

T = 2π ω

Differential Equation for SHM: d2x dt2 = − k m

  • x
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SLIDE 64

Newton’s Gravity April 29, 2013 - p. 10/12

General Solution

x = A cos (ωt + φ) A −A

T = 2π ω

Differential Equation for SHM: d2x dt2 = − k m

  • x

v = dx dt = −ωA sin (ωt + φ)

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SLIDE 65

Newton’s Gravity April 29, 2013 - p. 10/12

General Solution

x = A cos (ωt + φ) A −A

T = 2π ω

Differential Equation for SHM: d2x dt2 = − k m

  • x

v = dx dt = −ωA sin (ωt + φ) d2x dt2 = −ω2A cos (ωt + φ)

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SLIDE 66

Newton’s Gravity April 29, 2013 - p. 10/12

General Solution

x = A cos (ωt + φ) A −A

T = 2π ω

Differential Equation for SHM: d2x dt2 = − k m

  • x

v = dx dt = −ωA sin (ωt + φ) d2x dt2 = −ω2A cos (ωt + φ) = −ω2x

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SLIDE 67

Newton’s Gravity April 29, 2013 - p. 10/12

General Solution

x = A cos (ωt + φ) A −A

T = 2π ω

Differential Equation for SHM: d2x dt2 = − k m

  • x

v = dx dt = −ωA sin (ωt + φ) d2x dt2 = −ω2A cos (ωt + φ) = −ω2x

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SLIDE 68

Newton’s Gravity April 29, 2013 - p. 10/12

General Solution

x = A cos (ωt + φ) A −A

T = 2π ω

Differential Equation for SHM: d2x dt2 = − k m

  • x

v = dx dt = −ωA sin (ωt + φ) d2x dt2 = −ω2A cos (ωt + φ) = −ω2x ω2 = k m

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Newton’s Gravity April 29, 2013 - p. 10/12

General Solution

x = A cos (ωt + φ) A −A

T = 2π ω

Differential Equation for SHM: d2x dt2 = − k m

  • x

v = dx dt = −ωA sin (ωt + φ) d2x dt2 = −ω2A cos (ωt + φ) = −ω2x ω2 = k m

  • ω =
  • k

m

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Newton’s Gravity April 29, 2013 - p. 11/12

SHM Exercise

Three position-versus-time graphs are shown. Assuming the mass was the same for each, in which case was the spring constant largest?

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SLIDE 71

Newton’s Gravity April 29, 2013 - p. 11/12

SHM Exercise

Three position-versus-time graphs are shown. Assuming the mass was the same for each, in which case was the spring constant largest? (a) t x

2 4 6 8

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SLIDE 72

Newton’s Gravity April 29, 2013 - p. 11/12

SHM Exercise

Three position-versus-time graphs are shown. Assuming the mass was the same for each, in which case was the spring constant largest? (a) t x

2 4 6 8

(b) t x

2 4 6 8

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SLIDE 73

Newton’s Gravity April 29, 2013 - p. 11/12

SHM Exercise

Three position-versus-time graphs are shown. Assuming the mass was the same for each, in which case was the spring constant largest? (a) t x

2 4 6 8

(b) t x

2 4 6 8

(c) t x

2 4 6 8

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SLIDE 74

Newton’s Gravity April 29, 2013 - p. 11/12

SHM Exercise

Three position-versus-time graphs are shown. Assuming the mass was the same for each, in which case was the spring constant largest? (a) t x

2 4 6 8

(b) t x

2 4 6 8

(c) t x

2 4 6 8

(d) The spring constant is the same for each

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SLIDE 75

Newton’s Gravity April 29, 2013 - p. 11/12

SHM Exercise

Three position-versus-time graphs are shown. Assuming the mass was the same for each, in which case was the spring constant largest? (a) t x

2 4 6 8

(b) t x

2 4 6 8

(c) t x

2 4 6 8

(d) The spring constant is the same for each (e) There is not enough information to determine

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SLIDE 76

Newton’s Gravity April 29, 2013 - p. 11/12

SHM Exercise

Three position-versus-time graphs are shown. Assuming the mass was the same for each, in which case was the spring constant largest? (a) t x

2 4 6 8

(b) t x

2 4 6 8

(c) t x

2 4 6 8

(d) The spring constant is the same for each (e) There is not enough information to determine

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SLIDE 77

Newton’s Gravity April 29, 2013 - p. 11/12

SHM Exercise

Three position-versus-time graphs are shown. Assuming the mass was the same for each, in which case was the spring constant largest? (a) t x

2 4 6 8

T = 2π ω and ω =

  • k

m ⇒ T = 2π m k So the largest spring constant would have the shortest period (and largest frequency)

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SLIDE 78

Newton’s Gravity April 29, 2013 - p. 12/12

SHM Exercise II

Three position-versus-time graphs are shown. Assuming the spring constant was the same for each, in which case was the mass largest?

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SLIDE 79

Newton’s Gravity April 29, 2013 - p. 12/12

SHM Exercise II

Three position-versus-time graphs are shown. Assuming the spring constant was the same for each, in which case was the mass largest? (a) t x

2 4 6 8

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SLIDE 80

Newton’s Gravity April 29, 2013 - p. 12/12

SHM Exercise II

Three position-versus-time graphs are shown. Assuming the spring constant was the same for each, in which case was the mass largest? (a) t x

2 4 6 8

(b) t x

2 4 6 8

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SLIDE 81

Newton’s Gravity April 29, 2013 - p. 12/12

SHM Exercise II

Three position-versus-time graphs are shown. Assuming the spring constant was the same for each, in which case was the mass largest? (a) t x

2 4 6 8

(b) t x

2 4 6 8

(c) t x

2 4 6 8

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SLIDE 82

Newton’s Gravity April 29, 2013 - p. 12/12

SHM Exercise II

Three position-versus-time graphs are shown. Assuming the spring constant was the same for each, in which case was the mass largest? l (a) t x

2 4 6 8

(b) t x

2 4 6 8

(c) t x

2 4 6 8

(d) The mass is the same for each

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SLIDE 83

Newton’s Gravity April 29, 2013 - p. 12/12

SHM Exercise II

Three position-versus-time graphs are shown. Assuming the spring constant was the same for each, in which case was the mass largest? (a) t x

2 4 6 8

(b) t x

2 4 6 8

(c) t x

2 4 6 8

(d) The mass is the same for each (e) There is not enough information to determine

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SLIDE 84

Newton’s Gravity April 29, 2013 - p. 12/12

SHM Exercise II

Three position-versus-time graphs are shown. Assuming the spring constant was the same for each, in which case was the mass largest? (a) t x

2 4 6 8

(b) t x

2 4 6 8

(c) t x

2 4 6 8

(d) The mass is the same for each (e) There is not enough information to determine

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SLIDE 85

Newton’s Gravity April 29, 2013 - p. 12/12

SHM Exercise II

Three position-versus-time graphs are shown. Assuming the spring constant was the same for each, in which case was the mass largest? (c) t x

2 4 6 8

T = 2π m k So the largest mass would have the longest period (and smallest frequency)