University Physics: Waves and Electricity Ch15. Simple Harmonic Motion Lecture 2 Dr.-Ing. Erwin...

University Physics: Waves and Electricity

Ch15. Simple Harmonic MotionLecture 2

Dr.-Ing. Erwin Sitompulhttp://zitompul.wordpress.com

2/2Erwin Sitompul University Physics: Waves and Electricity

Plot the following simple harmonic motions in three different plots:(a) xa(t) = xmcosωt (b) xb(t) = xmcos(ωt–π/2) (c) xc(t) = xm/2cos(ωt+π/2)(d) xd(t) = 2xmcos(2ωt+π)

Homework 1: Plotting the Motions


Plot the following simple harmonic motions:(a) xa(t) = xmcosωt (b) xb(t) = xmcos(ωt–π/2) (c) xc(t) = xm/2cos(ωt+π/2)(d) xd(t) = 2xmcos(2ωt+π)

xa(t)

T0.5T

xm

–xm

0

xb(t)

xa(t)

T0.5T

xm

–xm

0xc(t)

xm/2

–xm/2

Solution of Homework 1: Plotting the Motions


Plot the following simple harmonic motions:(a) xa(t) = xmcosωt (b) xb(t) = xmcos(ωt–π/2) (c) xc(t) = xm/2cos(ωt+π/2)(d) xd(t) = 2xmcos(2ωt+π)

xa(t)

T0.5T

xm

–xm

0

xd(t)

2xm

–2xm

Solution of Homework 1: Plotting the Motions


Now we can use Newton’s second law to learn what force must act on the particle to give it that acceleration:

2( )F ma m x 2( )F m x

This result – a restoring force that is proportional to the displacement but opposite in sign – is familiar.

It is Hooke’s law for a spring:F kx

with the spring constant being k = mω2.

Conclusion: Simple harmonic motion is the motion executed by a particle, subject to a force that is proportional to the displacement of the particle but opposite in sign.

The Force Law for Simple Harmonic Motion


The block–spring system below forms a linear simple harmonic oscillator (or linear oscillator, for short).

The relation between the angular frequency ω of the simple harmonic motion of the block is related to the spring constant k and the mass m of the block as follows:

k

m

For the period of the linear oscillator, after combining, we can write:

2m

Tk

The Force Law for Simple Harmonic Motion


A block whose mass m is 680 g is fastened to a spring whose spring constant k is 65 N/m. The block is pulled a distance x = 11 cm from its equilibrium position at x = 0 on a frictionless surface and released from rest at t = 0.(a) What are the angular frequency, the frequency, and the period of the resulting motion?

(b) What is the amplitude of the oscillation?

(c) What is the maximum speed vm of the oscillating block, and where is the block when it has this speed?

k

m 65 N m

0.68 kg 9.78 rad s,

11 cmmx

m mv x (9.78)(0.11) 1.08 m s• Where is the block?

1.557 Hz,f 0.642 sT

Example 1


A block whose mass m is 680 g is fastened to a spring whose spring constant k is 65 N/m. The block is pulled a distance x = 11 cm from its equilibrium position at x = 0 on a frictionless surface and released from rest at t = 0.

(d) What is the magnitude am of the maximum acceleration of the block?

(e) What is the phase constant Φ for the motion?

(f) What is the displacement function x(t) for the spring–block system?

2m ma x 2(9.78) (0.11) 210.52 m s

0 rad

( ) cos( )mx t x t 0.11 m cos(9.78 ), in secondst t

• Why?

Example 1


At t = 0, the displacement x(0) of the block in a linear oscillator is –8.5 cm. The block’s velocity v(0) the is –0.920 m/s, and its acceleration a(0) is +47 m/s2.

(0) cosmx x

2(0) cosma x

(0)

(0)

a

x (0) sinmv x

47

( 0.085)

23.51 rad s

(a) What is the angular frequency ω of this system?

Example 2


(b) What are the phase constant Φ and amplitude xm?(0) sin

tan(0) cos

v

x

(0)tan

(0)

v

x

( 0.920)

(23.51)( 0.085)

0.460

24.7 or 24.7 180 155.3

At t = 0, the displacement x(0) of the block in a linear oscillator is –8.5 cm. The block’s velocity v(0) the is –0.920 m/s, and its acceleration a(0) is +47 m/s2.

? ?

Since(0) 0x cos 0 155.3

(0)

cosm

xx

0.085

cos155.3

0.0936 m 9.36 cm

Example 2


A block with m = 500 g is attached to a massless spring and is free to oscillate on a frictionless floor. The equilibrium position is 25 cm from the origin of the x axis.At t = 0, the block is pressed 10 cm from the equilibrium and then released. A force of magnitude 5 N is needed to press the spring.

k m

(a) How many oscillations occur in 1 minute?

F kx Fk

x

5 N

( 0.1 m)

50 N m

(50 N m) (0.5 kg) 10rad s

2f 10oscillations s

2 300

oscillations minute

95.493 oscillations minute

Example 3


(b) Determine the block’s equation of motion.

(c) Plot the motion of the block for two periods.

equil( ) cos( )mx t x x t equil 0.25 mx 0.1 mmx

(0) 0.15 mx • How?(0) 0.15 0.25 0.1cosx

cos 1,

( ) 0.25 0.1cos(10 ), in m, in sx t t x t

x(t)

T0.5T

0.25+xm

0.25–xm

00.252T1.5T

Example 3

University Physics: Waves and Electricity

Ch16. Transverse WavesLecture 2

Dr.-Ing. Erwin Sitompulhttp://zitompul.wordpress.com


Waves are of three main types:1.Mechanical waves. Common examples include water waves,

sound waves, and seismic waves. These waves are governed by Newton's laws and they can exist only within a material medium such as water, air, and rock.

2.Electromagnetic waves. Common examples include visible and ultraviolet light, radio and television waves, microwaves, x rays, and radar waves. These waves require no material medium to exist; for example light waves.

3.Matter waves. These waves are associated with electrons, protons, and other fundamental particles, and even atoms and molecules.

Much of what we discuss here applies to all kinds of waves. But for specific examples we shall refer to mechanical waves.

Types of Waves


A wave sent along a stretched, taut string is the simplest mechanical wave.

A single up-and-down motion generates a single pulse. The pulse moves along the string at some velocity v.

Assumptions:• No friction-like forces within the string, where the energy of the

wave may dissipate → No energy loss• Strings are very long → No need to consider reflected waves from

the far end

• A typical string element (marked with a dot) moves up once and then down as the pulse passes.

• The element’s motion is perpendicular to the wave’s direction of travel → the pulse is a transverse wave

→

A Simple Mechanical Wave


If a sinusoidal wave is sent along the string, we would find that the displacement of every string element is perpendicular to the direction of travel of the wave. (Figure on the left)

This motion is said to be transverse, and the wave is said to be a transverse wave.

If you push and pull on a piston in simple harmonic motion, a sinusoidal sound wave travels along the pipe. (Figure on the right)

Because the motion of the elements of air is parallel to the direction of the wave’s travel, the motion is said to be longitudinal, and the wave is said to be a longitudinal wave.

Transverse and Longitudinal Waves


Both a transverse wave and a longitudinal wave are said to be traveling wave because they both travel from one point to another.

Note that it is the wave that moves from one end to another end, not the material (string or air) through which the wave moves.

Transverse and Longitudinal Waves

• Longitudinal wave

• Transverse wave


To completely describe a wave on a string (and the motion of any element along its length), we need a function that gives the shape of the wave.

This means, we need a relation in the form y = f(x,t),

in which y is the transverse displacement of any string element as a function f of the time t and the position x of the element along the string.

If the wave is sinusoidal, then at time t, the displacement y of the element located at position x is given by:

( , ) sin( )my x t y kx t

Wavelength and Frequency


Traveling wave:• Five snapshots of a sinusoidal wave traveling in

the positive right direction of an x axis is shown in the figure.

• The movement of the wave is indicated by the short arrow pointing to a peak of the wave.

• But, at the meantime, the string moves only parallel to the x axis. Examine the motion of the red-colored string element at x = 0.

Wavelength and Frequency


The wavelength λ of a wave is the distance (parallel to the direction of the wave’s travel) between repetitions of the wave shape.

The angular wave number k is related to the wavelength by:

2k

The SI unit of k is radian per meter or m–1. Differentiate this k with the one that represents a spring

constant as previously, where the unit is N/m.

Wavelength and Angular Wave Number


The period of oscillation T of a wave is the time any string element takes to move through one full oscillation.

The angular frequency ω is related to the period by:2

T

The SI unit of ω is radian per second or s–1. The frequency f of a wave is defined as 1/T and is related to

the angular frequency ω by:1

2fT

Period, Angular Frequency, and Frequency


The figure is a composite of three snapshots, each of a wave traveling along a particular string.The phases for the waves are given by:(a) 2x–4t(b) 4x–8t(c) 8x–16tWhich phase corresponds to which wave in the figure?

(a) ↔ 2

(b) ↔ 3

(c) ↔ 1

Checkpoint


Two sinusoidal traveling waves with identical amplitude, wavelength, and period, can also be shifted one from another.

To accommodate this, the wave function can be generalized by inserting a phase constant Φ:


• Now, estimate the phase constant of this wave.

5

Phase Constant


The figure shows two snapshots of the sinusoidal wave, taken a small time interval Δt apart.

The wave is traveling in the positive direction of x (to the right), and the entire wave pattern moves a distance Δx during the time interval Δt.

After some derivation, we can write the wave speed v as:

vk

T

f

The Speed of a Traveling Wave




The equation that describes a wave moving in the positive direction is:


The equation that describes a wave moving in the negative direction is:


Here are the equations of three waves:(1) y(x,t) = 2sin(4x–2t)(2) y(x,t) = sin(3x–4t)(3) y(x,t) = 2sin(3x–3t)Rank the waves according to their (a) wave speed and (b) maximum speed perpendicular to the wave’s direction of travel (the transverse speed), greatest first.

(a) Wave speed: 2, 3, 1

Wave speed: vk

(b) Transverse speed: 3, then 1 and 2 tie

Transverse speed: m mv x



A wave traveling along a string is described byy(x,t) = 0.00327sin(72.1x–2.72t),

in which the numerical constants are in SI units.(a) What is the amplitude of this wave?

(b) What are the wavelength, period, and frequency of this wave?

0.00327 m 3.27 mmmy

2k

2

k

2

72.1

0.0871 m 8.71 cm

2

T

2

T

2

2.72

2.310 s

1fT

1

2.310 0.433 Hz

Example 1


(c) What is the velocity of this wave?

(d) What is the displacement y at x = 22.5 cm and t = 18.9 s?

A wave traveling along a string is described byy(x,t) = 0.00327sin(72.1x–2.72t),

in which the numerical constants are in SI units.

vk

2.72

72.1 0.0377 m s 3.77 cm s

( , ) 0.00327sin(72.1 2.72 )y x t x t (0.225 m,18.9 s) 0.00327sin(72.1 0.225 2.72 18.9)y

0.00327sin( 35.1855 rad) (0.00327 m)(0.588)0.00192 m

1.92 mm

ˆ3.77i cm sv

• How come?

Example 1


A sinusoidal wave of frequency 500 Hz has a speed of 350 m/s. (a) How far apart are two points that differ in phase by π/3 rad?(b) What is the phase difference between two displacements

at a certain point at times 1 ms apart?

Homework 2: Phase Differences

A sinusoidal wave of wavelength 280 cm has a speed of 60 m/s. (a) How many seconds apart are two displacements at a certain point that differ in phase by 2π/3 rad?(b) What is the phase difference between two displacements

at a certain time at distances 14 cm apart?

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