PHYSICS 231INTRODUCTORY PHYSICS I
PHYSICS 231INTRODUCTORY PHYSICS I
Lecture 21
Simple Harmonic Motion
, f, T determined by mass and spring constant
A, determined by initial conditions: x(0), v(0)
Last Lecture
Example 13.3
A 36-kg block is attached to a spring of constant k=600N/m. The block is pulled 3.5 cm away from itsequilibrium position and is pushed so that is has an initialvelocity 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 theexpression,
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
Example 13.4b
An object undergoing simple harmonic motion follows theexpression,
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
Example 13.4c
An object undergoing simple harmonic motion follows theexpression,
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
Example 13.4d
An object undergoing simple harmonic motion follows theexpression,
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
Example 13.4e
An object undergoing simple harmonic motion follows theexpression,
The object will pass through the equilibrium positionat 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
Simple Pendulum
Looks like Hooke’s law (k mg/L)
Simple Pendulum
Simple pendulum
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 roofalmost 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 thefree-fall acceleration is 1.67 m/s2, what is the periodof the pendulum there?
a) 59.7 m
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
wavelength
x
Snapshot of Longitudinal Wave
y could refer to pressure or density
Moving Wave
moves to right with velocity v
Fixing x=0,
Replace x with x-vtif wave moves to the right.Replace with x+vt ifwave should move to left.
Moving Wave: Formula Summary
- moving to right
+ moving to left
Example 13.6a
A wave traveling in thepositive x direction has afrequency of f = 25.0 Hz asshown in the figure. Thewavelength is:
a) 5 cm
b) 9 cm
c) 10 cm
d) 18 cm
e) 20 cm
Example 13.6b
A wave traveling in thepositive x direction has afrequency of f = 25.0 Hz asshown in the figure. Theamplitude is:
a) 5 cm
b) 9 cm
c) 10 cm
d) 18 cm
e) 20 cm
Example 13.6c
A wave traveling in thepositive x direction has afrequency of f = 25.0 Hz asshown in the figure. The speedof the wave is:
a) 25 cm/s
b) 50 cm/s
c) 100 cm/s
d) 250 cm/s
e) 500 cm/s
Example 13.7a
Consider the following expression for a pressure wave,where it is assumed that x is in cm,t is in seconds andP 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
Consider the following expression for a pressure wave,where it is assumed that x is in cm,t is in seconds andP 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
Consider the following expression for a pressure wave,where it is assumed that x is in cm,t is in seconds andP 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) 3 Hz
Example 13.7d
Consider the following expression for a pressure wave,where it is assumed that x is in cm,t is in seconds andP 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
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
Speed of a Wave in a VibratingString
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 Bas a communication device. If one wants todouble 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
Superposition Principle
Traveling waves can passthrough each other withoutbeing altered.
Reflection – Fixed End
Reflected wave is inverted
Reflection – Free End
Reflected pulse not inverted