Chapter 13 Vibrations and Waves. When x is positive, F is negative ; When at equilibrium (x=0), F = 0 ; When x is negative, F is positive ; Hooke’s Law.

Presentation on theme: "Chapter 13 Vibrations and Waves. When x is positive, F is negative ; When at equilibrium (x=0), F = 0 ; When x is negative, F is positive ; Hooke’s Law."— Presentation transcript:

1 Chapter 13 Vibrations and Waves

2 When x is positive, F is negative ; When at equilibrium (x=0), F = 0 ; When x is negative, F is positive ; Hooke’s Law Reviewed

3 Sinusoidal Oscillation Pen traces a sine wave

4 Graphing x vs. t A : amplitude (length, m)T : period (time, s) A T

5 Some Vocabulary f = Frequency  = Angular Frequency T = Period A = Amplitude  = phase

6 Phases Phase is related to starting time 90-degrees changes cosine to sine

7 a x v Velocity is 90  “out of phase” with x: When x is at max, v is at min.... Acceleration is 180° “out of phase” with x a = F/m = - (k/m) x Velocity and Acceleration vs. time T T T

8 v and a vs. t Find v max with E conservation Find a max using F=ma

9 What is  ? Requires calculus. Since

10 Formula Summary

11 Example13.1 An block-spring system oscillates with an amplitude of 3.5 cm. If the spring constant is 250 N/m and the block has a mass of 0.50 kg, determine (a) the mechanical energy of the system (b) the maximum speed of the block (c) the maximum acceleration. a) 0.153 J b) 0.783 m/s c) 17.5 m/s 2

12 Example 13.2 A 36-kg block is attached to a spring of constant k=600 N/m. The block is pulled 3.5 cm away from its equilibrium positions and released from rest at t=0. At t=0.75 seconds, a) what is the position of the block? b) what is the velocity of the block? a) -3.489 cm b) -1.138 cm/s

13 Example 13.3 A 36-kg block is attached to a spring of constant k=600 N/m. The block is pulled 3.5 cm away from its equilibrium position and is pushed so that is has an initial velocity of 5.0 cm/s at t=0. a) What is the position of the block at t=0.75 seconds? a) -3.39 cm

14 Example 13.4a An object undergoing simple harmonic motion follows the expression, The amplitude of the motion is: a) 1 cm b) 2 cm c) 3 cm d) 4 cm e) -4 cm Where x will be in cm if t is in seconds

15 Example 13.4b An object undergoing simple harmonic motion follows the expression, The period of the motion is: a) 1/3 s b) 1/2 s c) 1 s d) 2 s e) 2/  s Here, x will be in cm if t is in seconds

16 Example 13.4c An object undergoing simple harmonic motion follows the expression, The frequency of the motion is: a) 1/3 Hz b) 1/2 Hz c) 1 Hz d) 2 Hz e)  Hz Here, x will be in cm if t is in seconds

17 Example 13.4d An object undergoing simple harmonic motion follows the expression, The angular frequency of the motion is: a) 1/3 rad/s b) 1/2 rad/s c) 1 rad/s d) 2 rad/s e)  rad/s Here, x will be in cm if t is in seconds

18 Example 13.4e An object undergoing simple harmonic motion follows the expression, The object will pass through the equilibrium position at the times, t = _____ seconds a) …, -2, -1, 0, 1, 2 … b) …, -1.5, -0.5, 0.5, 1.5, 2.5, … c) …, -1.5, -1, -0.5, 0, 0.5, 1.0, 1.5, … d) …, -4, -2, 0, 2, 4, … e) …, -2.5, -0.5, 1.5, 3.5, Here, x will be in cm if t is in seconds

19 Simple Pendulum Looks like Hooke’s law (k  mg/L)

20 Simple Pendulum

21 Simple pendulum Frequency independent of mass and amplitude! (for small amplitudes)

22 Pendulum Demo

23 Example 13.5 A man enters a tall tower, needing to know its height h. He notes that a long pendulum extends from the roof almost to the ground and that its period is 15.5 s. (a) How tall is the tower? (b) If this pendulum is taken to the Moon, where the free-fall acceleration is 1.67 m/s 2, what is the period of the pendulum there? a) 59.7 m b) 37.6 s

24 Damped Oscillations In real systems, friction slows motion

25 TRAVELING WAVES Sound Surface of a liquid Vibration of strings Electromagnetic Radio waves Microwaves Infrared Visible Ultraviolet X-rays Gamma-rays Gravity

26 Longitudinal (Compression) Waves Sound waves are longitudinal waves

27 Compression and Transverse Waves Demo

28 Transverse Waves Elements move perpendicular to wave motion Elements move parallel to wave motion

29 Snapshot of a Transverse Wave wavelength x

30 Snapshot of Longitudinal Wave y could refer to pressure or density

31 Moving Wave moves to right with velocity v Fixing x=0, Replace x with x-vt if wave moves to the right. Replace with x+vt if wave should move to left.

32 Moving Wave: Formula Summary

33 Example 13.6a A wave traveling in the positive x direction has a frequency of f = 25.0 Hz as shown in the figure. The wavelength is: a) 5 cm b) 9 cm c) 10 cm d) 18 cm e) 20 cm

34 Example 13.6b A wave traveling in the positive x direction has a frequency of f = 25.0 Hz as shown in the figure. The amplitude is: a) 5 cm b) 9 cm c) 10 cm d) 18 cm e) 20 cm

35 Example 13.6c A wave traveling in the positive x direction has a frequency of f = 25.0 Hz as shown in the figure. The speed of the wave is: a) 25 cm/s b) 50 cm/s c) 100 cm/s d) 250 cm/s e) 500 cm/s

36 Example 13.7a Consider the following expression for a pressure wave, where it is assumed that x is in cm,t is in seconds and P will be given in N/m 2. What is the amplitude? a) 1.5 N/m 2 b) 3 N/m 2 c) 30 N/m 2 d) 60 N/m 2 e) 120 N/m 2

37 Example 13.7b Consider the following expression for a pressure wave, where it is assumed that x is in cm,t is in seconds and P will be given in N/m 2. What is the wavelength? a) 0.5 cm b) 1 cm c) 1.5 cm d)  cm e) 2  cm

38 Example 13.7c Consider the following expression for a pressure wave, where it is assumed that x is in cm,t is in seconds and P will be given in N/m 2. What is the frequency? a) 1.5 Hz b) 3 Hz c) 3/  Hz d) 3/(2  ) Hz e) 3  Hz

39 Example 13.7d Consider the following expression for a pressure wave, where it is assumed that x is in cm,t is in seconds and P will be given in N/m 2. What is the speed of the wave? a) 1.5 cm/s b) 6 cm/s c) 2/3 cm/s d) 3  /2 cm/s e) 2/  cm/s

40 Example 13.8 Which of these waves move in the positive x direction? a) 5 and 6 b) 1 and 4 c) 5,6,7 and 8 d) 1,4,5 and 8 e) 2,3,6 and 7

41 Speed of a Wave in a Vibrating String For different kinds of waves: (e.g. sound) Always a square root Numerator related to restoring force Denominator is some sort of mass density

42 Example 13.9 A string is tied tightly between points A and B as a communication device. If one wants to double the wave speed, one could: a) Double the tension b) Quadruple the tension c) Use a string with half the mass d) Use a string with double the mass e) Use a string with quadruple the mass

43 Superposition Principle Traveling waves can pass through each other without being altered.

44 Reflection – Fixed End Reflected wave is inverted

45 Reflection – Free End Reflected pulse not inverted