Oscillations and Simple Harmonic Motion : Mechanics C
Slide 2
Oscillatory Motion Oscillatory Motion is repetitive back and
forth motion about an equilibrium position Oscillatory Motion is
periodic. Swinging motion and vibrations are forms of Oscillatory
Motion. Objects that undergo Oscillatory Motion are called
Oscillators.
Slide 3
Simple Harmonic Motion The time to complete one full cycle of
oscillation is a Period. The amount of oscillations per second is
called frequency and is measured in Hertz.
Slide 4
What is the oscillation period for the broadcast of a 100MHz FM
radio station? Heinrich Hertz produced the first artificial radio
waves back in 1887!
Slide 5
Simple Harmonic Motion The most basic of all types of
oscillation is depicted on the bottom sinusoidal graph. Motion that
follows this pattern is called simple harmonic motion or SHM.
Slide 6
Simple Harmonic Motion An objects maximum displacement from its
equilibrium position is called the Amplitude (A) of the
motion.
Slide 7
What shape will a velocity-time graph have for SHM? Everywhere
the slope (first derivative) of the position graph is zero, the
velocity graph crosses through zero.
Slide 8
We need a position function to describe the motion above.
Slide 9
Mathematical Models of SHM x(t) to symbolize position as a
function of time A=x max =x min When t=T, cos(2 )=cos(0)
x(t)=A
Slide 10
Mathematical Models of SHM In this context we will call omega
Angular Frequency What is the physical meaning of the product (A)?
The maximum speed of an oscillation!
Slide 11
Recall: Hookes Law Here is what we want to do: DERIVE AN
EXPRESSION THAT DEFINES THE DISPLACEMENT FROM EQUILIBRIUM OF THE
SPRING IN TERMS OF TIME. WHAT DOES THIS MEAN? THE SECOND DERIVATIVE
OF A FUNCTION THAT IS ADDED TO A CONSTANT TIMES ITSELF IS EQUAL TO
ZERO. What kind of function will ALWAYS do this?
Slide 12
Slide 13
Example: An airtrack glider is attached to a spring, pulled
20cm to the right, and released at t=0s. It makes 15 oscillations
in 10 seconds. What is the period of oscillation?
Slide 14
Example: An airtrack glider is attached to a spring, pulled
20cm to the right, and released at t=0s. It makes 15 oscillations
in 10 seconds. What is the objects maximum speed?
Slide 15
Example: An airtrack glider is attached to a spring, pulled
20cm to the right, and released at t=0s. It makes 15 oscillations
in 10 seconds. What are the position and velocity at t=0.8s?
Slide 16
Example: A mass oscillating in SHM starts at x=A and has period
T. At what time, as a fraction of T, does the object first pass
through 0.5A?
Slide 17
Model of SHM When collecting and modeling data of SHM your
mathematical model had a value as shown below: What if your clock
didnt start at x=A or x=-A? This value represents our initial
conditions. We call it the phase angle: This value represents our
initial conditions. We call it the phase angle:
Slide 18
SHM and Circular Motion Uniform circular motion projected onto
one dimension is simple harmonic motion.
Slide 19
SHM and Circular Motion Start with the x-component of position
of the particle in UCM End with the same result as the spring in
SHM! Notice it started at angle zero
Slide 20
Initial conditions: We will not always start our clocks at one
amplitude.
Slide 21
The Phase Constant: Phi is called the phase of the oscillation
Phi naught is called the phase constant or phase shift. This value
specifies the initial conditions. Different values of the phase
constant correspond to different starting points on the circle and
thus to different initial conditions
Slide 22
Phase Shifts:
Slide 23
An object on a spring oscillates with a period of 0.8s and an
amplitude of 10cm. At t=0s, it is 5cm to the left of equilibrium
and moving to the left. What are its position and direction of
motion at t=2s? Initial conditions: From the period we get:
Slide 24
An object on a spring oscillates with a period of 0.8s and an
amplitude of 10cm. At t=0s, it is 5cm to the left of equilibrium
and moving to the left. What are its position and direction of
motion at t=2s?
Slide 25
We have modeled SHM mathematically. Now comes the physics.
Total mechanical energy is conserved for our SHM example of a
spring with constant k, mass m, and on a frictionless surface. The
particle has all potential energy at x=A and x=A, and the particle
has purely kinetic energy at x=0.
Slide 26
At turning points: At x=0: From conservation: Maximum speed as
related to amplitude:
Slide 27
From energy considerations: From kinematics: Combine
these:
Slide 28
a 500g block on a spring is pulled a distance of 20cm and
released. The subsequent oscillations are measured to have a period
of 0.8s. at what position or positions is the blocks speed 1.0m/s?
The motion is SHM and energy is conserved.
Slide 29
Dynamics of SHM Acceleration is at a maximum when the particle
is at maximum and minimum displacement from x=0.
Slide 30
Dynamics of SHM Acceleration is proportional to the negative of
the displacement.
Slide 31
Dynamics of SHM As we found with energy considerations:
According to Newtons 2 nd Law: Acceleration is not constant: This
is the equation of motion for a mass on a spring. It is of a
general form called a second order differential equation.
Slide 32
2 nd -Order Differential Equations: Unlike algebraic equations,
their solutions are not numbers, but functions. In SHM we are only
interested in one form so we can use our solution for many objects
undergoing SHM. Solutions to these diff. eqns. are unique (there is
only one). One common method of solving is guessing the solution
that the equation should have From evidence, we expect the
solution:
Slide 33
2 nd -Order Differential Equations: Lets put this possible
solution into our equation and see if we guessed right! IT WORKS.
Sinusoidal oscillation of SHM is a result of Newtons laws!
Slide 34
Vertical springs oscillate differently than horizontal springs
because there are 2 forces acting. The equilibrium position gets
shifted downward
Slide 35
What about vertical oscillations of a spring-mass system??
Hanging at rest: this is the equilibrium position of the
system.
Slide 36
Now we let the system oscillate. At maximum: But: So:
Everything that we have learned about horizontal oscillations is
equally valid for vertical oscillations!
Slide 37
You need to show how to derive the Period of a Pendulum
equation T = 2l/g
Slide 38
The Pendulum Equation of motion for a pendulum
Slide 39
Small Angle Approximation: When is about 0.1rad or less, h and
s are about the same.
Slide 40
The Pendulum Equation of motion for a pendulum
Slide 41
A Pendulum Clock What length pendulum will have a period of
exactly 1s?
Slide 42
Conditions for SHM Notice that all objects that we look at are
described the same mathematically. Any system with a linear
restoring force will undergo simple harmonic motion around the
equilibrium position.
Slide 43
A Physical Pendulum when there is mass in the entire pendulum,
not just the bob. Small Angle Approx.
Slide 44
Damped Oscillations All real oscillators are damped
oscillators. these are any that slow down and eventually stop. a
model of drag force for slow objects: b is the damping constant
(sort of like a coefficient of friction).
Slide 45
Damped Oscillations Another 2 nd -order diff eq. Solution to 2
nd - order diff eq:
Slide 46
Damped Oscillations A slowly changing line that provides a
border to a rapid oscillation is called the envelope of the
oscillations.
Slide 47
Slide 48
Driven Oscillations Not all oscillating objects are disturbed
from rest then allowed to move undisturbed. Some objects may be
subjected to a periodic external force.
Slide 49
Driven Oscillations All objects have a natural frequency at
which they tend to vibrate when disturbed. Objects may be exposed
to a periodic force with a particular driving frequency. If the
driven frequency matches the natural frequency of an object,
RESONANCE occurs
Slide 50
Spring Best example of simple harmonic oscillator. T = 2 m/k
m
Slide 51
Simple Pendulum Acts as simple harmonic oscillator only when
angle of swing is small. T = 2 L/g
Slide 52
Conical Pendulum Not really a simple harmonic oscillator, but
equation is similar to simple pendulum. T = 2 L(cos )/g
Slide 53
The small angle approximation for a simple pendulum mg mgcos
mgsin A simple pendulum is one where a mass is located at the end
of string. The strings length represents the radius of a circle and
has negligible mass. Once again, using our sine function model we
can derive using circular motion equations the formula for the
period of a pendulum. If the angle is small, the radian value for
theta and the sine of the theta in degrees will be equal.
Slide 54
Torsional Pendulum Twists back and forth through equilibrium
position. T = 2 I/
Slide 55
Physical Pendulum Anything that doesnt fall into any of the
other categories of pendulums. T = 2 I/ = Mgd
Slide 56
K 1 + U 1 = K 2 + U 2 K = 1 / 2 mv 2 U = mgh 1 / 2 mv 1 2 +mgh
1 = 1 / 2 mv 2 2 + mgh 2 v 1 2 + 2gh 1 = v 2 2 + 2gh 2 Energy
Conservation in Pendulums h
Slide 57
The Physical Pendulum A physical pendulum is an oscillating
body that rotates according to the location of its center of mass
rather than a simple pendulum where all the mass is located at the
end of a light string. It is important to understand that d is the
lever arm distance or the distance from the COM position to the
point of rotation. It is also the same d in the Parallel Axes
theorem.
Slide 58
Example A spring is hanging from the ceiling. You know that if
you elongate the spring by 3.0 meters, it will take 330 N of force
to hold it at that position: The spring is then hung and a 5.0-kg
mass is attached. The system is allowed to reach equilibrium; then
displaced an additional 1.5 meters and released. Calculate the:
Spring Constant Angular frequency 110 N/m 4.7 rad/s
Slide 59
Example A spring is hanging from the ceiling. You know that if
you elongate the spring by 3.0 meters, it will take 330 N of force
to hold it at that position: The spring is then hung and a 5.0-kg
mass is attached. The system is allowed to reach equilibrium; then
displaced an additional 1.5 meters and released. Calculate the:
Amplitude Frequency and Period Stated in the question as 1.5 m 0.75
Hz 1.34 s
Slide 60
Example A spring is hanging from the ceiling. You know that if
you elongate the spring by 3.0 meters, it will take 330 N of force
to hold it at that position: The spring is then hung and a 5.0-kg
mass is attached. The system is allowed to reach equilibrium; then
displaced an additional 1.5 meters and released. Calculate the:
Total Energy Maximum velocity 123.75 J 7.05 m/s
Slide 61
Position of mass at maximum velocity Maximum acceleration of
the mass Position of mass at maximum acceleration At the
equilibrium position 33.135 m/s/s At maximum amplitude, 1.5 m A
spring is hanging from the ceiling. You know that if you elongate
the spring by 3.0 meters, it will take 330 N of force to hold it at
that position: The spring is then hung and a 5.0-kg mass is
attached. The system is allowed to reach equilibrium; then
displaced an additional 1.5 meters and released. Calculate
the:
Slide 62
a)Find the length of a simple seconds pendulum. b)What
assumption have you made in this calculation? A seconds pendulum
beats seconds; that is, it takes 1 s for half a cycle.
Slide 63
Problem #2 A thin uniform rod of mass 0.112 kg and length 0.096
m is suspended by a wire through its center and perpendicular to
its length. The wire is twisted and the rod set to oscillating. The
period is found to be 2.14 s. When a flat body in the shape of an
equilateral triangle is suspended similarly through its center of
mass, the period is 5.83. Find the rotational inertia of the
triangle.
Slide 64
Problem #3 A uniform disk is pivoted at its rim. Find the
period for small oscillations and the length of the equivalent
simple pendulum.
