Elements of Waves and Thermal Physics Wed. 14:50 – 16:20 Place: Room 2214 Assoc. Prof. H. SAIBI, West building 2, 431, Ito Campus

Textbooks 2

Oscillations and Wave H. SAIBI October 8 th, 2014

NameUnit  Phase constant (delta)radians  ThetaAngle in degrees  PhiAngle in degrees T Period (T=2π/  ) second ttimesecond fFrequency (f=1/T)Cycle per second (cy/s) or Hertz (Hz) mmasskg (  t+  ) Phase of motion FxFx Linear restoring forceDynes in m (grams) and a x in cm/s 2. k Force constant of the spring or spring stiffness or spring force Newtons per metre  Angular frequencyRadians/second  Angular speed or angle of velocityRadians/second vxvx velocitycm/s axax accelerationcm/s 2 Aamplitudecm EMechanical energyJoul if A in m., T in sec., m in kg. IMoment of inertia  Angular acceleration (alpha)Radians/second 2 kTorsional constant  Torque (moment of force) (Tau)Newton-meter ggravitym/s 2 4

Outline  Simple Harmonic Motion  Energy in Simple Harmonic Motion  Some Oscillating Systems 5

Simple Harmonic Motion Position x vs. time t Definition of period T Definition of amplitude A

Phase Phase (Time)Phase difference

Frequency and Period Period (T) in seconds (s) and Frequency (f) in Hertz (Hz). Frequency range of human hearing (20 to 20,000 Hz): f = 20 Hz T = 0.05 s = 50 ms f = 20,000 Hz = 20 kHz T = 0.000,05 s = 0.05 ms = 50 ms

Very basic kind of oscillatory motion is: Simple Harmonic Motion. Hooke’s Law: … (1) (restoring force) Newton’s Second Law: … (2) Combination of (1) and (2): Simple Harmonic Motion Force constant In simple harmonic motion, the acceleration, and thus the net force, are both proportional to, and oppositely directed from, the displacement from the equilibrium position. …(3) …(4) Amplitude 9

Frequency (reciprocal of the period T): … (5) (cycle/second or Hz) T is the time it takes for a displaced object to execute a complete cycle of oscillatory motion. Position in simple harmonic motion: … (6) For info: … (7) System in phase: phase difference is ZERO or integer times 2 π. System 180 ° out of phase: phase difference is π or an odd integer times of π. … (8) … (9) Simple Harmonic Motion Phase of motion Phase constant 10

Simple Harmonic Motion Velocity in simple harmonic motion: … (10) Acceleration is: … (11) … (12) Comparing Eq. 12 with Eq. 4, we see that Eq. 6 is a solution of Eq. 4. if: The amplitude A and the phase constant can be determined from the initial position x 0 and the initial velocity v 0x of the system. Setting t=0 in … (13) ; Similarly, setting: … (14) gives: … (15) By these equations we can determine A and  in terms of x 0, v ox, and . 2 Acceleration in simple harmonic motion 11

Simple Harmonic Motion The period T is the shortest time interval satisfying the relation: … (16) For all t. Substituting into this relation using: … (17) … (18) The cosine (and sine) function repeats in value when the phase increases by 2 π, so: … (19) The constant  is called the angular frequency. It has the unit of radians per second and dimensions of inverse time, the same as angular speed. Substituting 2 π/ T for  in gives: … (20) The frequency is related to the angular frequency by: … (21) Because of … (22) “the frequency increases with increasing k and decreases with increasing mass” 12

Simple Harmonic Motion Figure: Two identical mass-spring systems. 13

Simple Harmonic Motion Figure: Plots of x versus t for the systems in Fig. Both reach their equilibrium positions at the same time. The frequency (and thus the period) of simple harmonic motions is independent of the amplitude 14

Simple Harmonic Motion If the phase constant  is zero, Equations 6, 10, and 11 then become: … (23) … (24) … (25) These functions are plotted in this Figure 15

Simple Harmonic Motion and Circular Motion A relation exists between simple harmonic motion and circular motion with constant speed. The angular displacement relative to the +x direction is given by: … (26) Angular speed of the particle is: … (27) The x component of the particle’s position is: … (28) “simple harmonic motion” … (29) Angular displacement at t=0 When a particle moves with constant speed in a circle, its projection onto a diameter of the circle moves with simple harmonic motion. Same as Eq. 10 (simple harmonic motion). 16

Energy in Simple Harmonic Motion When an object on a spring undergoes simple harmonic motion, the system’s potential energy and kinetic energy vary with time. Their sum, the total mechanical energy is constant: … (30) The potential energy is: … (31) The kinetic energy of the system is: … (32) The total mechanical energy E is: … (33) The total mechanical energy in simple harmonic motion is proportional to the square of the amplitude 17

Energy in Simple Harmonic Motion Fig. Plots of x, U, and K versus t. Fig. The potential energy function. The average value of the potential energy and the kinetic energy over one or more cycles are equal, their average values are given by: … (34) Turning points 18

General Motion Near Equilibrium Simple harmonic motion occurs when a particle is displaced slightly from a position of stable equilibrium. The equation for a parabola that has a minimum at point x 1 is: … (35) The force is related to the potential energy curve by: … (36) Figure: Plot of U vs. x. A and B are constants. Unstable equilibrium Stable equilibrium A Stable equilibrium at x=0 19

Some Oscillating Systems – Object on a vertical spring - The net force of the spring is: … (37) If we set y’=y-y0 in Eq. (37) we get: … (38) … (39) … (40) Newton’s Second Law: … 20

Some Oscillating Systems – Object on a vertical spring - is the same Eq. (4) with y’ replacing x. It has now the familiar solution: where: (angular frequency)  A force is conservative if the work done by it is independent of the path. Both the force of spring and the force of gravity are conservative, and the sum of these forces (Eqs. 37 & 40) also is conservative.  The potential energy function U associated with the sum of these forces is the negative of the work done plus an arbitrary integration constant, that is: … (41) 21

The Simple Pendulum bob string String tension weight Arc length The mass m does not appear in Eq.(43), the motion of a pendulum does not depend on its mass. For small  and … (44) The mass m does not appear in Eq.(43), the motion of a pendulum does not depend on its mass. For small  and … (44) Period: Using tangential components, Newton’s Second Law ( ) gives: …(42) (s: arc length) Differentiating both sides of: gives:  Rearranging gives: … (43) Eq.(44) is the same form with Eq.(4). Thus, the motion of a pendulum approximates simple harmonic motion for small angular displacements. Eq.(44) can be written: …(45) and …(46) (period of a simple pendulum for small oscillations) 22

The Simple Pendulum The solution of the Eq.(45) is: where  0 is the maximum angular displacement. According to Eq.(46) : the greater the length of a pendulum, the greater the period. – The period and therefore the frequency are independent of the amplitude of oscillation (as long as the amplitude is small). This statement is a general feature of simple harmonic motion. We can measure gravity “g” using a simple pendulum undergoing small oscillations. We need only to measure the length l and period T of the pendulum, and using Eq.(46), solve for “g”. Lab. Experiment on Oct. 25 th,

Pendulum in an accelerated reference frame The clock keeps time by using a torsional oscillator (a) Simple pendulum in apparent equilibrium. (b) in accelerated frame. a  0 is the equilibrium angle and Applying Newton’s second law to the bob gives: If the bob remains at rest relative to the boxcar, then: and If the bob is moving relative to the boxcar, then: (a’ is the acceleration of the bob relative to the boxcar). Substituting for a in Eq.(47) gives: … (47) Some rearrangement 24

Large-Amplitude Oscillations Fig. T/To vs. Amplitude. Period for large-amplitude oscillations When the amplitude of a pendulum’s oscillation becomes large, its motion continues to be periodic, but it is no longer a simple harmonic. For an angular amplitude  0, the period can be shown to be given by: where: 25

The Torsional Oscillator A system that undergoes rotational oscillations in a variation of simple-harmonic motion is called a torsional oscillator. and Eq.(49) is similar to Eq.(4) and its solution is: where is the angular frequency (NOT the angular speed) of the motion. The period is therefore: Fig. The torsional pendulum consists of a solid disk suspended by a steel wire. Fig. Mechanical clock (period of oscillation is constant = keep time). Torsional constant of the wire. Newton’s second law for rotational motion Angular acceleration …(48) …(49) …(50) Period of a torsional oscillator …(51) 26

The Physical Pendulum Fig. A physical Pendulum. A rigid object free to rotate about a horizontal axis that is not through its center of mass will oscillate when displaced from equilibrium. Such system is called a physical pendulum. Comparing Eq.(52) with Eq.(48), we can see that for small angular displacements the physical pendulum is a torsional oscillator with a torsional constant given by: Thus the motion of the pendulum is described by Eq.(50) with. The period is therefore: … (52) Period of a physical pendulum depends on distribution of the mass. … (53) Moment of Inertia 27