Lecture 2 Ch 15 Simple Harmonic Motion University

Lecture 2 Ch 15. Simple Harmonic Motion University Physics: Waves and Electricity Dr. -Ing. Erwin Sitompul http: //zitompul. wordpress. com

Homework 1: Plotting the Motions Plot the following simple harmonic motions in three different plots: (a) xa(t) = xm cosωt (b) xb(t) = xm cos(ωt–π/2) (c) xc(t) = xm/2 cos(ωt+π/2) (d) xd(t) = 2 xm cos(2ωt+π) Erwin Sitompul University Physics: Waves and Electricity 2/2

Solution of Homework 1: Plotting the Motions Plot the following simple xm harmonic motions: (a) xa(t) = xm cosωt 0 (b) xb(t) = xm cos(ωt–π/2) (c) xc(t) = xm/2 cos(ωt+π/2) –xm (d) xd(t) = 2 xm cos(2ωt+π) xm xm/2 0 –xm/2 –xm Erwin Sitompul xb(t) xa(t) 0. 5 T T xc(t) University Physics: Waves and Electricity 2/3

Solution of Homework 1: Plotting the Motions Plot the following simple harmonic motions: 2 xm (a) xa(t) = xm cosωt xm (b) xb(t) = xm cos(ωt–π/2) (c) xc(t) = xm/2 cos(ωt+π/2) 0 (d) xd(t) = 2 xm cos(2ωt+π) –xm xa(t) 0. 5 T T xd(t) – 2 xm Erwin Sitompul University Physics: Waves and Electricity 2/4

The Force Law for Simple Harmonic Motion § Now we can use Newton’s second law to learn what force must act on the particle to give it that acceleration: § 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: 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. Erwin Sitompul University Physics: Waves and Electricity 2/5

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: § For the period of the linear oscillator, after combining, we can write: Erwin Sitompul University Physics: Waves and Electricity 2/6

Example I 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? • Where is the block? Erwin Sitompul University Physics: Waves and Electricity 2/7

Example I 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? • Why? (f) What is the displacement function x(t) for the spring–block system? Erwin Sitompul University Physics: Waves and Electricity 2/8

Example II 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/s 2. (a) What is the angular frequency ω of this system? Erwin Sitompul University Physics: Waves and Electricity 2/9

Example II 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/s 2. (b) What are the phase constant Φ and amplitude xm? ? ? Since Erwin Sitompul University Physics: Waves and Electricity 2/10

Example III 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. (a) How many oscillations occur in 1 minute? Erwin Sitompul University Physics: Waves and Electricity 2/11

Example III (b) Determine the block’s equation of motion. • How? (c) Plot the motion of the block for two periods. 0. 25+xm x(t) 0. 25 0 0. 5 T T 1. 5 T 2 T 0. 25–xm Erwin Sitompul University Physics: Waves and Electricity 2/12

Lecture 2 Ch 16. Transverse Waves University Physics: Waves and Electricity Dr. -Ing. Erwin Sitompul http: //zitompul. wordpress. com

Types of 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. Erwin Sitompul University Physics: Waves and Electricity 2/14

A Simple Mechanical Wave § 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. • 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 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 Erwin Sitompul University Physics: Waves and Electricity 2/15

Transverse and Longitudinal Waves § 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. Erwin Sitompul University Physics: Waves and Electricity 2/16

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. Erwin Sitompul University Physics: Waves and Electricity 2/17

Wavelength and Frequency § 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: Erwin Sitompul University Physics: Waves and Electricity 2/18

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. Erwin Sitompul University Physics: Waves and Electricity 2/19

Wavelength and Angular Wave Number § 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: § 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. Erwin Sitompul University Physics: Waves and Electricity 2/20

Period, Angular Frequency, and Frequency § 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: § 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: Erwin Sitompul University Physics: Waves and Electricity 2/21

Checkpoint 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) 2 x– 4 t (b) 4 x– 8 t (c) 8 x– 16 t Which phase corresponds to which wave in the figure? (a) ↔ 2 (b) ↔ 3 (c) ↔ 1 Erwin Sitompul University Physics: Waves and Electricity 2/22

Phase Constant § Two sinusoidal traveling waves with identical amplitude, wavelength, and period, can also be shifted one from another. • Now, estimate the phase constant of this wave. § To accommodate this, the wave function can be generalized by inserting a phase constant Φ: Erwin Sitompul University Physics: Waves and Electricity 2/23

The Speed of a Traveling Wave § 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: Erwin Sitompul University Physics: Waves and Electricity 2/24

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: Erwin Sitompul University Physics: Waves and Electricity 2/25

Checkpoint Here are the equations of three waves: (1) y(x, t) = 2 sin(4 x– 2 t) (2) y(x, t) = sin(3 x– 4 t) (3) y(x, t) = 2 sin(3 x– 3 t) 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 (b) Transverse speed: 3, then 1 and 2 tie Wave speed: Transverse speed: Erwin Sitompul University Physics: Waves and Electricity 2/26

Example I A wave traveling along a string is described by y(x, t) = 0. 00327 sin(72. 1 x– 2. 72 t), 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? Erwin Sitompul University Physics: Waves and Electricity 2/27

Example I A wave traveling along a string is described by y(x, t) = 0. 00327 sin(72. 1 x– 2. 72 t), in which the numerical constants are in SI units. (c) What is the velocity of this wave? • How come? (d) What is the displacement y at x = 22. 5 cm and t = 18. 9 s? Erwin Sitompul University Physics: Waves and Electricity 2/28

Homework 2: Phase Differences 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? New 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? Erwin Sitompul University Physics: Waves and Electricity 2/29