University Physics: Waves and Electricity. Ch15. Simple Harmonic Motion. Lecture 2. Dr.-Ing. Erwin Sitompul. http://zitompul.wordpress.com. 2013. Homework 1: Plotting the Motions. Plot the following simple harmonic motions in three different plots: (a) x a ( t ) = x m cos ωt
University Physics: Waves and Electricity Ch15. Simple Harmonic Motion Lecture 2 Dr.-Ing. Erwin Sitompul http://zitompul.wordpress.com 2013
Homework 1: Plotting the Motions 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)
Solution of Homework 1: Plotting the Motions xm xm 0 0 0.5T 0.5T T T –xm –xm 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) xb(t) xa(t) xm/2 xa(t) xc(t) –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) 2xm xm xa(t) 0 0.5T T –xm xd(t) –2xm
Object on a Vertical Spring • When an object hangs from a vertical spring, there is a force mg downward in addition to the force of the spring. • If we choose the upward direction to be positive, the spring’s force on the object is Fs = ky, with y as the difference in the stretched and unstretched position of the spring. • The Newton’s 2nd Law gives:
Object on a Vertical Spring • After some manipulations, the solution can be found as
Energy in Simple Harmonic Motion • The energy of a linear oscillator transfers back and forth between kinetic energy and potential energy, while the sum of the two –the mechanical energy E of the oscillator– remains constant. • The potential energy is associated entirely with the spring. Its value depends on how much the spring is stretched or compressed. • The kinetic energy of the system is associated entirely with the block. Its value depends on how fast the block is moving.
Energy in Simple Harmonic Motion • The mechanical energy is then given by • The mechanical energy of a linear oscillator is indeed constant and independent of time. • The potential energy and kinetic energy of a linear oscillator are shown next as the function of time and as the function of displacement.
Simple Harmonic Motion and Uniform Circular Motion • Simple harmonic motion is actually the projection of uniform circular motion on a diameter of the circle in which the circular motion occurs. • Measurement of the angle between Jupiter and its moon Callisto as seen from Earth.
x(t) v(t) Simple Harmonic Motion and Uniform Circular Motion • Next we can see, whether we look at the displacement, the velocity, or the acceleration, the projection of uniform circular motion is indeed simple harmonic motion.
Damped Simple Harmonic Motion • When the motion of an oscillator is reduced by an external force, the oscillator and its motion are said to be damped. • An idealized example of a damped oscillator is shown on the next figure. • As the vane moves up and down, the liquid exerts an inhibiting drag force on it and thus on the entire oscillating system. • With time, the mechanical energy of the block-spring system decreases, as energy is transferred to thermal energy of the liquid and vane.
Damped Simple Harmonic Motion • Let us assume the liquid exerts a damping force Fd that is proportional to the velocity v of the vane and block. Then, for the components along the x axis → → where b is a damping constant, that depends on both the vane and the liquid, with the SI unit of kilogram/s. • The force on the block from the spring is Fs= –kx. Let us assume that the gravitational force on the block is taken out of calculation by setting the position where the mass rests as the equilibrium position of the system.
Damped Simple Harmonic Motion • Then we can write Newton’s 2nd Law for the components along the x axis as • The solution of this equation is where xmis the amplitude and ω’ is the angular frequency of the damped oscillation. This angular frequency is given by
Damped Simple Harmonic Motion • The solution can be seen as a cosine function whose amplitude, which is xme–bt/2m, gradually decreases with time, as the next figure suggests. • If the damping is small, we can find the mechanical energy by replacing xmas found in the formula previously with xme–bt/2m, so that
Checkpoint Here are three sets of values for the spring constant, damping constant, and mass for the damped oscillator. Rank the sets according to the time required for the mechanical energy to decrease to one-fourth of its initial value, greatest first. • The greater the magnitude of the exponential, the faster the decrease. Set 1, Set 2, then Set 3
Example 1 For the damped oscillator, m=250 g, k=85 N/m, and b=70 g/s. (a) What is the period of the motion? (b) How long does it take for the amplitude of the damped oscillation to drop to half its initial value? • Or after around 15 periods of oscillation
Example 1 For the damped oscillator, m=250 kg, k=85 N/m, and b=70 g/s. (c) How long does it take for the mechanical energy to drop to one-half its initial value? • Auto Shock- Absorber
University Physics: Waves and Electricity Ch16. Transverse Waves Lecture 2 Dr.-Ing. Erwin Sitompul http://zitompul.wordpress.com
Types of Waves Waves are of three main types: 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. 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. 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.
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
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.
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 wave • Longitudinal wave
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:
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 Angular Wave Number • The wavelengthλ 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.
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:
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) 2x–4t (b) 4x–8t (c) 8x–16t Which phase corresponds to which wave in the figure? (a) ↔2 (b) ↔ 3 (c) ↔ 1
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:
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:
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:
The Speed of a Traveling Wave 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 (b) Transverse speed: 3, then 1 and 2 tie Wave speed: Transverse speed:
Example 1 A wave traveling along a string is described by y(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?
Example 1 A wave traveling along a string is described by y(x,t) = 0.00327sin(72.1x–2.72t), 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?
Email Quiz A hungry scorpion can detect the presence of a nearby beetle by the waves the motion sends along the sand surface while the beetle moves. The waves are of two types: transverse wave with the speed of vt = 50 m/s and longitudinal waves traveling at vl= 140 m/s. If a careless motion of the beetle sends out such waves, the scorpion can determine the distance of the beetle from the difference Δt in the arrival times of the waves at its leg nearest to the beetle. If Δt is given as 4.0 ms, determine the beetle’s distance d (in cm).
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?
Homework 2A: Phase Differences 1. 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? 2. In the figure below, two springs with the same value of spring constant 7580 N/m are connected to a box with the mass 0.245 kg. Determine the oscillation frequency on the frictionless floor?