Periodic MotionPeriodic MotionSimple periodic motionSimple periodic motion is that motion in is that motion in which a body moves back and forth over a which a body moves back and forth over a fixed path, returning to each position and fixed path, returning to each position and velocity after a definite interval of time.velocity after a definite interval of time.

AmplitudeA

PeriodPeriod, T, is the time for one complete oscillation. (seconds,s)(seconds,s)

PeriodPeriod, T, is the time for one complete oscillation. (seconds,s)(seconds,s)

FrequencyFrequency, f, is the number of complete oscillations per second. Hertz (sHertz (s-1-1))

FrequencyFrequency, f, is the number of complete oscillations per second. Hertz (sHertz (s-1-1))

1f

T

Example 1:Example 1: The suspended mass The suspended mass makes 30 complete oscillations in 15 s. makes 30 complete oscillations in 15 s. What is the period and frequency of the What is the period and frequency of the motion?motion?

x FF

15 s0.50 s

30 cylcesT

Period: T = 0.500 sPeriod: T = 0.500 s

1 1

0.500 sf

T Frequency: f = 2.00 HzFrequency: f = 2.00 Hz

Simple Harmonic Motion, Simple Harmonic Motion, SHMSHM

Simple harmonic motionSimple harmonic motion is periodic motion in is periodic motion in the absence of friction and produced by a the absence of friction and produced by a restoring force that is directly proportional to restoring force that is directly proportional to the displacement and oppositely directed.the displacement and oppositely directed.

A restoring force, F, acts in the direction opposite the displacement of the oscillating body.

F = -kx

A restoring force, F, acts in the direction opposite the displacement of the oscillating body.

F = -kx

x FF

Hooke’s LawHooke’s LawWhen a spring is stretched, there is a

restoring force that is proportional to the displacement.

F = -kx

The spring constant k is a property of the spring given

by:

k = F

x

F

x

m

Displacement in SHMDisplacement in SHM

m

x = 0 x = +Ax = -A

x

• Displacement is positive when the position is to the right of the equilibrium position (x = 0) and negative when located to the left.

• The maximum displacement is called the amplitude A.

Period and Frequency as a Period and Frequency as a Function of Mass and Spring Function of Mass and Spring

Constant.Constant.For a vibrating body with an For a vibrating body with an elastic restoring elastic restoring force:force:

Recall that Recall that F = ma = -kxF = ma = -kx:

1

2

kf

m

1

2

kf

m 2

mT

k2

mT

k

The frequency f and the period T can be found if the spring constant k and mass m of the vibrating body are known. Use consistent SI units.

The frequency f and the period T can be found if the spring constant k and mass m of the vibrating body are known. Use consistent SI units.

The Simple PendulumThe Simple Pendulum

The period of a The period of a simple simple pendulumpendulum is given by: is given by:

mg

L

2L

Tg

For small angles

1

2

gf

L

SummarySummary

Simple harmonic motion (SHM)Simple harmonic motion (SHM) is that is that motion in which a body moves back and motion in which a body moves back and forth over a fixed path, returning to each forth over a fixed path, returning to each position and velocity after a definite position and velocity after a definite interval of time.interval of time.

Simple harmonic motion (SHM)Simple harmonic motion (SHM) is that is that motion in which a body moves back and motion in which a body moves back and forth over a fixed path, returning to each forth over a fixed path, returning to each position and velocity after a definite position and velocity after a definite interval of time.interval of time.

1f

TF

x

m

The frequency (rev/s) is the reciprocal of the period (time for one revolution).

The frequency (rev/s) is the reciprocal of the period (time for one revolution).

Summary (Cont.)Summary (Cont.)

F

x

m

Hooke’s Law: In a spring, there is a restoring force that is proportional to the displacement.

Hooke’s Law: In a spring, there is a restoring force that is proportional to the displacement.

The spring constant k is defined by:

Fk

x

Fk

x

F kxF kx

Summary (SHM)Summary (SHM)

F ma kx F ma kx kxa

m

kxa

m

m

x = 0 x = +Ax = -A

x va

½mvA2 + ½kxA 2 = ½mvB

2 + ½kxB 2 ½mvA2 + ½kxA 2 = ½mvB

2 + ½kxB 2

Conservation of Energy:

Summary (SHM)Summary (SHM)

2 2kv A x

m

2 2kv A x

m

2 2 21 1 12 2 2mv kx kA 2 2 21 1 1

2 2 2mv kx kA

0

kv A

m0

kv A

m

cos(2 )x A ft cos(2 )x A ft

2 sin(2 )v fA ft 2 sin(2 )v fA ft

2 24a f x 2 24a f x

Summary: Period and Summary: Period and Frequency for Vibrating Frequency for Vibrating

Spring.Spring.m

x = 0 x = +Ax = -A

x va

1

2

af

x

1

2

af

x

2x

Ta

2

xT

a

2m

Tk

2m

Tk

1

2

kf

m

1

2

kf

m

Summary: Simple Summary: Simple Pendulum and Torsion Pendulum and Torsion

PendulumPendulum

2L

Tg

1

2

gf

L

L

2'

IT

k