Chapter 10

Simple Harmonic Motion

10.1 The Ideal Spring and Simple Harmonic Motion

HOOKE’S LAW: RESTORING FORCE OF AN IDEAL SPRING

The restoring force on an ideal spring is xkFx −=

K is called theSpring constant.Or spring stiffness constant

The stiffer the spring, the greaterThe spring constant.

The motion of the mass is periodic.

The mass vibrates or oscillatesAbout its equilibrium position.

The force is not constant so theAcceleration is not constant.

Robert Hooke Newton “If I have seen a little further it is by standing on the shoulders of Giants."

1) A 12 cm long spring has a force constant (k) of 400N/m . How much force required to stretch the spring to a length of 14cm ?

2) When a family of four with a total mass of 200 kg step into their1200 kg car, the car's spring compress 3.0cm.A) What is the spring constant of the car's springs, assumingThey act as a single spring ?

B) How far will the car lower if loaded with a 300kg rather than 200kg /

1. F = kx = 400 (0.02) = 8N2. k=6.5E4 N/mx=4.5R-2m

Simple Harmonic Oscillations, KINEMATICS. Vocabulary:

Displacement = distance x of the mass from equilibrium

Amplitude=The maximum displacement from the equilibrium point

One cycle= refers to the complete to-and-fro motion from some initial point

The period T = the time required to complete a complete cycle

The frequency f =number of cycles per second in hertz (Hz)1 Hz= 1 cycle per second with f = 1/TExample: 10 cycles in 1 second means f = 1 Hz and T = 0.1s

Any vibrating system for which the restoring force is proportional to the negativeOf the displacement (F=-Kx or F = -kΘ) is said to exhibit simple harmonic motion (SHM). Example: Is the motion simple harmonic?F=-0.5x2 ? F = -2.3x ? f = 8.6x ? F = -4Θ ?

GO BACK TO THE applet.

What happens to the period when the mass increases ?What happens to the period when the spring constant K Increases ?

1. A block oscillating on the end of a spring moves from its position ofmaximum spring compression in 0.25s. Determine the period and frequencyOf this motion. (f=1/T)

2. A student observing an oscillating block counts 45 cycles of oscillation in one Minute. Determine its frequency (in hertz) and period (in second)

3. How would be the period of the spring-block oscillator change if bothThe mass of the block and the spring constant were doubles ?(T = 2 π sqrt(m/K) )

4. A block is attached to a spring and set into oscillatory motion, and its frequency is Measured. If this block were removed and replaced by a second block with ¼ the mass of the first block. How would the frequency of the oscillations compared to that of the fist block ?

5. A spider of mass 0.3g waits in its web of negligible mass . A slight movement

Causes the web to vibrate with a frequency of about 15 Hz.

A) Estimate the value of the spring stiffness constant for the web.

B) At what frequency would you expect the web to vibrate if an insect of mass 0.1g

Were trapped in addition to the spider ?

K=2.7 N/mF=13 Hz

KINEMATICS for SHM

Remember last semester ?Displacement versus time was:X = vt if the speed was constant , acceleration = 0, the graph x(t) is a ....

X = Vot + 0.5 a t2 if acceleration is constant, the graph x(t) is a ....

RUN THE applet motion of a spring.

The graph x(t) is ..... ?

Play with your TI to find the equation x(t).Suppose that t=0, then x(t) = A. so is it sine or cosine ?Suppose A = 1, Plot x(t) = cos(wt) with w =1, 2, 3 .. How the constant wChanges the graph ? w depends on what ? (see above?)

So w is inversely proportional to the period w= 2π / T ,

X = A cos ( wt)

period T: the time required to complete one cycle

frequency f: the number of cycles per second (measured in Hz)

Tf 1=

Tf ππω 22 ==

amplitude A: the maximum displacement

tAvvv

Tx ωωθ sinsinmax

−=−=

SO NOW we need to find the speed and the acceleration.Observe the applet again.

What happens to the speed and acceleration ?

Acceleration = force / massSo when force = 0, then acceleration = 0

Some calculus:

Speed = dx/dt and acceleration = dv/dt

tAax ωω cos2−=

1. The displacement of an object is described by the following equation, where xIs in meter and t is in seconds:

x=0.3m cos(8 t )

Determine the oscillating object:A) amplitude

B) frequency

C) period

D) Maximum speed

E) Maximum acceleration

X = A cos ( wt) with w = 2π /TV=-Aw sin(wt) and a = -Aw2 cos(wt)

A=0.3m , B) f=1.27Hz, C) T=0.79s D) Vmax=2.4m/s E) a=19m/s/s

10.2 Simple Harmonic Motion and the Reference Circle

Example 3 The Maximum Speed of a Loudspeaker Diaphragm

The frequency of motion is 1.0 KHz and the amplitude is 0.20 mm. (a)What is the maximum speed of the diaphragm?(b)Where in the motion does this maximum speed occur?

X = A cos ( wt) with w = 2π /TV=-Aw sin(wt) and a = -Aw2 cos(wt)

10.2 Simple Harmonic Motion and the Reference Circle

tAvvv

Tx ωωθ sinsinmax

−=−=

(a) ( ) ( )( ) ( )sm3.1

Hz100.12m1020.02 33max

=××=== − ππω fAAv

(a)The maximum speedoccurs midway betweenthe ends of its motion.

10.2 Simple Harmonic Motion and the Reference Circle

Example 6 A Body Mass Measurement Device

The device consists of a spring-mounted chair in which the astronautsits. The spring has a spring constant of 606 N/m and the mass ofthe chair is 12.0 kg. The measured period is 2.41 s. Find the mass of theastronaut.

kmT =

Π2

mk

T=Π2

kmT Π= 2

astrochairtotal mmm +=

10.2 Simple Harmonic Motion and the Reference Circle

( ) astrochair2total 2mm

Tkm +==

π

( )( )( ) kg 77.2kg 0.12

4s 41.2mN606

2

2

2

chair2astro

=−=

−=

π

πm

Tkm

The clone of a loudspeaker vibrates in SHM at a frequency of 262 Hz (“ middle C” ) . The amplitude of the center of the cone is A=1.5 E-4m, and atT=0, x=A.

A) What equation describes the motion of the center of the cone ?

B) What are the velocity and acceleration as a function of time ?

C) What is the position of the cone at t = 1ms ( 0.001s = E-3 s)Switch mode to radians.

10.3 Energy and Simple Harmonic Motion

DEFINITION OF ELASTIC POTENTIAL ENERGY

The elastic potential energy is the energy that a springhas by virtue of being stretched or compressed. For anideal spring, the elastic potential energy is

221 mvKE =

SI Unit of Elastic Potential Energy: joule (J)

221

elasticPE kx=

EM= KE + PEel

1. A block of mass 0.2kg on a spring whose force constant is k=500N/m.The amplitude is 4.0cm. Calculate the maximum speed of the block.

X=0KE is max, vmaxPE=0

X=XmaxKE=0PE is max , v=0

X=XmaxKE=0PE is max , v=0

2. A block of mass is attached to an ideal spring of force constant k = 500N/m/The amplitude of the resulting oscillations is 10cm.Determine the total energy of the oscillator and the kinetic energy of theOscillator and the kinetic energy of the block when it's 2cm from equilibrum.

1. Vmax = 2m/s2. E=2.5J and KE = 2.4J

3. A block of mass m= 2kg is attached to an ideal spring of spring constant k=200N/m . The block is at rest at its equilibrium position. An impulsive force acts on the block, giving it an initial speed of 2m/s. Find the amplitude of the resulting oscillations.

3. A= 0.2m

10.4 The Pendulum

A simple pendulum consists of a particle attached to a frictionlesspivot by a cable of negligible mass.

gL

Lg Π== 2Tor only) angles (small ω

10.4 The Pendulum

Example 10 Keeping Time

Determine the length of a simple pendulum that willswing back and forth in simple harmonic motion with a period of 1.00 s.

22Lg

Tf === ππω

2

2

4πgTL =

10.4 The Pendulum

Example 10 Keeping Time

Determine the length of a simple pendulum that willswing back and forth in simple harmonic motion with a period of 1.00 s.

22Lg

Tf === ππω

( ) ( ) m 248.04

sm80.9s 00.14 2

22

2

2

===ππ

gTL

2

2

4πgTL =

10.5 Damped Harmonic Motion

In simple harmonic motion, an object oscillated with a constant amplitude.

In reality, friction or some other energy dissipating mechanism is always present and the amplitude decreases as time passes.

This is referred to as damped harmonic motion.

10.5 Damped Harmonic Motion

1) simple harmonic motion

2&3) underdamped

1) critically damped

5) overdamped

10.6 Driven Harmonic Motion and Resonance

When a force is applied to an oscillating system at all times,the result is driven harmonic motion.

Here, the driving force has the same frequency as the spring system and always points in the direction of the object’s velocity.

10.6 Driven Harmonic Motion and Resonance

RESONANCE

Resonance is the condition in which a time-dependent force can transmitlarge amounts of energy to an oscillating object, leading to a large amplitudemotion.

Resonance occurs when the frequency of the force matches a natural frequency at which the object will oscillate.