PHYSICS 231 INTRODUCTORY PHYSICS I Lecture 21 Last Lecture Simple Harmonic Motion x = Acos(t ) v= Asin(t ) a = 2 A(cost ) k

= m 1 f = T 2 =2 = T , f, T f, f, T T f, T determined by mass and spring constant A, f, T determined by initial conditions: x(0), v(0) 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 Example 13.4a An object undergoing simple harmonic motion follows the expression,

x(t) =4 + 2 cos[ (t 3)] here x will be in cm if t is in seconds he amplitude of the motion is: ) 1 cm ) 2 cm ) 3 cm ) 4 cm ) -4 cm Example 13.4b An object undergoing simple harmonic motion follows the expression,

x(t) =4 + 2 cos[ (t 3)] ere, x will be in cm if t is in seconds The period of the motion is: a) 1/3 s b) 1/2 s c) 1 s d) 2 s e) 2/ s Example 13.4c An object undergoing simple harmonic motion

follows the expression, x(t) =4 + 2 cos[ (t 3)] ere, x will be in cm if t is in seconds The frequency of the motion is: a) 1/3 Hz b) 1/2 Hz c) 1 Hz d) 2 Hz e) Hz Example 13.4d

An object undergoing simple harmonic motion follows the expression, x(t) =4 + 2 cos[ (t 3)] ere, x will be in cm if t is in seconds he angular frequency of the motion is: ) 1/3 rad/s ) 1/2 rad/s ) 1 rad/s ) 2 rad/s ) rad/s

Example 13.4e An object undergoing simple harmonic motion follows the expression, x(t) =4 + 2 cos[ (t 3)] ere, x will be in cm if t is in seconds object will pass through the equilibrium position the times, t = _____ seconds , , ,

, , -2, -1, 0, 1, 2 -1.5, -0.5, 0.5, 1.5, 2.5, -1.5, -1, -0.5, 0, 0.5, 1.0, 1.5, -4, -2, 0, 2, 4, -2.5, -0.5, 1.5, 3.5, Simple Pendulum F =m g sin x x

sin = x2 + L2 L mg F x L Looks like Hookes law (k mg/L) Simple Pendulum F =m g sin x x

sin = x2 + L2 L mg F x L g = L = max cos(t ) Simple pendulum g

= L Frequency independent of mass and amplitude! (for small amplitudes) Pendulum Demo 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? a) 59.7 m

(b) If this pendulum is taken to the Moon, where the free-fall acceleration is 1.67 m/s2, what is the period of the pendulum there? b) 37.6 s Damped Oscillations In real systems, friction slows motion TRAVELING WAVES Sound

Surface of a liquid Vibration of strings Electromagnetic Radio waves Microwaves Infrared Visible Ultraviolet X-rays Gamma-rays Gravity Longitudinal (Compression) Waves

Elements move parallel to wave motion. Example - Sound waves Transverse Waves Elements move perpendicular to wave motion. Examples - strings, light waves Compression and Transverse Waves Demo Snapshot of a Transverse Wave

x y = Acos 2 wavelength x Snapshot of Longitudinal Wave

x y = Acos 2 y could refer to pressure or density Moving Wave y = Acos 2 Replace x with x-vt x vt if wave moves to the right

Replace with x+vt if wave should move to left. moves to right with velocity v Fixing x=0, v y = Acos 2 t

v f = , v= Moving Wave: Formula Summary v = x y = Acos2 m t

- moving to right + moving to left 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)

b) c) d) e) 5 cm 9 cm 10 cm 18 cm 20 cm 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) b) c) d) e) 5 cm 9 cm 10 cm 18 cm 20 cm

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) b) c) d) e)

25 cm/s 50 cm/s 100 cm/s 250 cm/s 500 cm/s Example 13.7a nsider the following expression for a pressure wave P =60 cos(2 x 3t) ere it is assumed that x is in cm,t is in seconds a will be given in N/m2.

What is the amplitude? a) 1.5 N/m2 b) 3 N/m2 c) 30 N/m2 d) 60 N/m2 e) 120 N/m2 Example 13.7b nsider the following expression for a pressure wave P =60 cos(2 x 3t) ere it is assumed that x is in cm,t is in seconds a

will be given in N/m2. What is the wavelength? a) 0.5 cm b) 1 cm c) 1.5 cm d) cm e) 2 cm Example 13.7c nsider the following expression for a pressure wave P =60 cos(2 x 3t)

ere it is assumed that x is in cm,t is in seconds a will be given in N/m2. What is the frequency? a) 1.5 Hz b) 3 Hz c) 3/ Hz d) 3/(2) Hz e) 32 Hz Example 13.7d nsider the following expression for a pressure wave P =60 cos(2 x 3t)

ere it is assumed that x is in cm,t is in seconds a will be given in N/m2. 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 Example 13.8 ich of these waves move in the positive x direction

1) y =21.3cos(3.4 x + 2.5t) 2) y=21.3cos(3.4 x 2.5t) 3) y=21.3cos(3.4 x + 2.5t) 4) y=21.3cos(3.4 x 2.5t) 5) y=21.3cos(3.4 x + 2.5t) 6) y=21.3cos(3.4 x 2.5t) 7) y=21.3cos(3.4 x + 2.5t) 8) y=21.3cos(3.4 x 2.5t) a) b) c) d) e)

5 and 1 and 5,6,7 1,4,5 2,3,6 6 4 and 8 and 8 and 7 Speed of a Wave in a Vibrating

String T m v= here m = m L For other kinds of waves: (e.g. sound) Always a square root Numerator related to restoring force Denominator is some sort of mass density 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) b) c) d) e) Double the tension Quadruple the tension Use a string with half the mass

Use a string with double the mass Use a string with quadruple the mass Superposition Principle Traveling waves can pass through each other without being altered. y(x,t) = y1 (x, t)+ y2 (x, t) Reflection Fixed End Reflected wave is inverted

Reflection Free End Reflected pulse not inverted