Chapter 12Chapter 12Simple Harmonic MotionSimple Harmonic Motion

Photo by Mark Tippens

A TRAMPOLINE exerts a restoring force on the jumper that is directly proportional to the average force required to displace the mat. Such restoring forces provide the driving forces necessary for objects that oscillate with simple harmonic motion.

After finishing this After finishing this section, you should be section, you should be able to:able to:

• Write and apply Write and apply Hooke’s LawHooke’s Law for objects for objects moving with simple harmonic motion.moving with simple harmonic motion.

• Describe the motion of Describe the motion of pendulumspendulums and calculate the and calculate the length length required to produce a required to produce a given given frequency.frequency.

• Write and apply formulas for Write and apply formulas for finding the finding the frequency frequency ff,, period period TT,, velocity velocity vv, or , or accelerationacceleration aa in terms of in terms of displacementdisplacement xx or or time time tt..

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

Hooke's law is the relationship Hooke's law is the relationship between the force exerted on between the force exerted on the mass and its position x the mass and its position x

Example 2:Example 2: A 4-kg mass suspended A 4-kg mass suspended from a spring produces a from a spring produces a displacement of 20 cm. What is the displacement of 20 cm. What is the spring constant?spring constant?

F20 cm

m

The stretching force is the The stretching force is the weight (W = mg) of the 4-kg weight (W = mg) of the 4-kg

mass:mass:

F = F = (4 kg)(9.8 m/s(4 kg)(9.8 m/s22) = 39.2 N) = 39.2 N

Now, from Hooke’s law, the force Now, from Hooke’s law, the force constant k of the spring is:constant k of the spring is:

k = =k = =

FF

xx

0.2 m0.2 mk = 196

N/mk = 196

N/m

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.

Example 3:Example 3: The frictionless system The frictionless system shown below has a shown below has a 2-kg2-kg mass mass attached to a spring (attached to a spring (k = 400 N/mk = 400 N/m). ). The mass is displaced a distance of The mass is displaced a distance of 20 cm20 cm to the right and released. to the right and released.What is the frequency of the motion?What is the frequency of the motion?

m

x = 0 x = +0.2 m

x va

x = -0.2 m

1 1 400 N/m

2 2 2 kg

kf

m

f = 2.25 Hzf = 2.25 Hz

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

Spring.Spring.

m

x = 0 x = +Ax = -A

x va

2m

Tk

2m

Tk

1

2

kf

m

1

2

kf

m

Summary: Simple Pendulum Summary: Simple Pendulum

2L

Tg

1

2

gf

L

L