Chapter 11Elasticity And Periodic Motion

Goals for Chapter 11

• To follow periodic motion to a study of simple harmonic motion.

• To solve equations of simple harmonic motion.

• To use the pendulum as a prototypical system undergoing simple harmonic motion.

PERIODIC MOTION

Periodic Motion

Fx = -kx

Using Newton’s second law,max = -kx

OR

ax = -(k/m)x

Note the linear relationship between ax and x

ax = -(k/m)x

• Amplitude, A

• Cycle

• Period, T

• Frequency, f f = 1/T SI unit : Hertz (Hz) = cycle/s = 1/s

• Angular frequency, ω ω = 2πf = 2π/T

Conservation of Mechanical EnergyE = (1/2) mvx

2 + (1/2)kx2 = constant

Energy in Simple Harmonic Motion

When x = ± A, vx = 0. At this point, the energy is entirely potential energy and E = (1/2)kA2 .

E = (1/2)kA2 = (1/2) mvx2 + (1/2)kx2

vx = ± k/m A2 – x2

We can use this equation to find the magnitude of the velocity for any given position x.

a) Force cst; b) vmax and vmin; c) amax and amin;d) v and a half-way to center; e) K, U and E in halfway

Equations of Simple Harmonic Motion

The relationship between uniform

circular motion and simple harmonic

motion.

x = A cos θ x = A cos(ωt) x = A cos [(2π/T )t]SI unit: m

ω is the angular frequency

Position of the Shadow as a Function of Time

Velocity in Simple Harmonic Motion

v = -Aω sin(ωt)SI unit: m/s

Maximum speed of the mass is vmax = Aω

Acceleration in Simple Harmonic Motion

a = -Aω2 cos(ωt)

SI unit: m/s2

Maximum acceleration has a magnitudeamax = Aω2

T = 2π m / kSI unit: s

Period of a Mass on a Spring

From equation ax = -Aω2 cos(ωt)ax = -Aω2 (maximum acceleration) (1)Also, we know that ax = -kx/m ax = -kA/m (maximum acceleration) (2)From equations (1) and (2)-Aω2 = -kx/mω2 = k/mω = k/m = 2πf = 2π/T

The Simple PendulumThe Simple Pendulum

The Simple Pendulum

The restoring force F at each point is the component of force tangent to the circular path at that point:

F = -mgsinθ

If the angle is small, sinθ is very nearly equal to θ (in radians).

F = -mgθ = -mgx/LF = -(mg/L)x

The restoring force F is then proportional to the coordinate x for small displacements, and the constant mg/L represents the force constant k.

CHAPTER 12MECHANICAL WAVES AND SOUND

Edvard MunchThe scream

A disturbance that propagates from one place to another is referred to as a wave.

Waves propagate with well-defined speeds determined by the properties of the material through which they travel.

Waves carry energy.

In a transverse wave individual particles move at right angles to the direction of wave propagation.

In a longitudinal wave individual particles move in the same direction as the wave propagation.

A wave on a string

As a wave on a string moves

horizontally, all points on the

string vibrate in the vertical direction.

Water waves from a disturbance.

Wavelength, Frequency, and

Speed

vvwavewave = = λλ /T /T λλ f = v f = vwavewave

Speed of a wave

REFLECTIONS AND SUPERPOSITION

A reflected wave pulse: fixed end

A reflected wave pulse: free end

The Principle of supperposition:

Whenever two waves overlap, the actualdisplacement of any point on the string,at any time, is obtained by vector additionof the following two displacements:

1)The displacement the point would have if ONLY the first wave were present

2) The displacement the point would have if ONLY the second wave were present

Constructive Interference

Destructive Interference

Figure 14-22Figure 14-22Interference with Two SourcesInterference with Two Sources

In phase/opposite phase: Two sources are in phase if they both emit crests at the same time. Sources have opposite phase if one emits a crest at the same time other emits a trough.

Constructive interference occurs when the path length from the two sources differs by 0, λ, 2λ, 3λ, …….

Destructive interference occurs when the path length from the two sources differs by λ/2, 3λ/2, 5λ/2, …….

Sound Waves

Speed of Sound in Air

v = 343 m/s

The frequency of sound determines its pitch. High-pitched sounds have high frequencies; low-pitched sounds have

low frequencies.

Human hearing extends from 20 Hz to 20, 000 Hz. Sounds with frequencies above this range are referred to as

ultrasonic, while those with frequencies lower than 20 Hz are classified as

infrasonic.

Waves become coherentWaves become coherentDepending on the shape and size of the medium transmitting Depending on the shape and size of the medium transmitting the wave, different standing wave patterns are established as a the wave, different standing wave patterns are established as a function of energy.function of energy.

Normal modes for a linear resonatorNormal modes for a linear resonator• The resonator is fixed at both ends.

• Wave energy increases as you go down the y axis below.

Fundamental frequenciesFundamental frequencies The fundamental frequency depends on the properties of the The fundamental frequency depends on the properties of the resonant medium. resonant medium. If the resonator is a string, cord, or wire, the standing wave If the resonator is a string, cord, or wire, the standing wave pattern is a function of tension, linear mass density, and length.pattern is a function of tension, linear mass density, and length.