last time: simple pendulum - santa rosa junior college
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LAST TIME: Simple Pendulum:
The displacement from equilibrium, x is the arclength s = L.
2L
g
sinF mg mg
/ /s L x L
xF mg ma
L
ga x
L
Accelerating & Restoring Force in the tangential
direction, taking cw as positive initial
displacement direction and assume small angle
approximation:
2x
g
L
STRATEGY:
If you can show that the system obeys Hooke’s Law:
Force ~ - Displacement
Then you get to ASSUME the system moves in SHM and that:
OR
So you simplify the equation down to this and whatever is the coefficient of –x is the square of ω, the angular frequency!!!
2a x 2
Physical Pendulum: Rods & Disks If a hanging object oscillates
about a fixed axis that does not pass through the center of mass and the object cannot be approximated as a particle, the system is called a physical pendulum It cannot be treated as a
simple pendulum The gravitational force provides
a torque about an axis through O
The magnitude of the torque is mgd sin
I is the moment of inertia about the axis through O
Physical Pendulm Sample Problem
1. A uniform thin rod (length L = 1.0 m, mass = 2.0 kg) is suspended from a pivot at one end. Assuming small oscillations, derive an expression for the angular frequency in terms of the given variables (m, L, g), and then solve for a numerical value in rad/s. Show all your work. Sketch a diagram showing angle, lengths, lever arms, etc, and explain whatever is needed for a fantastic solution.
© 2013 Pearson Education, Inc.
The Physical Pendulum
Any solid object that swings
back and forth under the
influence of gravity can be
modeled as a physical pendulum.
The gravitational torque for
small angles ( 10) is:
Plugging this into Newton’s second law for rotational
motion, I, we find the equation for SHM, with:
Slide 14-82
2. A uniform disk (R = 1.0 m, m = 2.0 kg) is suspended from a pivot a distance 0.25 m above its center of mass. Ignore air resistance and any other frictional forces. Starting from Newton’s Second Law and assuming small oscillations, derive a reduced expression for the angular frequency in terms of the given variables: (R, m, g), and then solve for a numerical value in rad/s. Show all your work. Sketch a diagram showing angle, lengths, lever arms, etc, and explain whatever is needed for a fantastic solution.
Physical Pendulm Sample Problem
QuickCheck 14.14
A. The solid disk.
B. The circular hoop.
C. Both have the same period.
D. There’s not enough information to tell.
A solid disk and a circular hoop
have the same radius and the
same mass. Each can swing back
and forth as a pendulum from a
pivot at one edge. Which has the
larger period of oscillation?
Slide 14-85
QuickCheck 14.14
A. The solid disk.
B. The circular hoop.
C. Both have the same period.
D. There’s not enough information to tell.
A solid disk and a circular hoop
have the same radius and the
same mass. Each can swing back
and forth as a pendulum from a
pivot at one edge. Which has the
larger period of oscillation?
Slide 14-86
A hoop made of a thin wire of mass M and radius R is pinned at its edge as shown. Find the period of oscillation.
© 2013 Pearson Education, Inc.
Damped Oscillations
Position-versus-time graph for a damped oscillator.
Slide 14-89
© 2013 Pearson Education, Inc.
Driven Oscillations and Resonance
F0 is the driving force
0 is the natural frequency of the undamped oscillator
b is the damping constant
The figure shows the same
oscillator with three different values
of the damping constant.
The resonance amplitude becomes
higher and narrower as the
damping constant decreases.
0
22
2 2
0
FmA
b
m
Resonance
Resonance (maximum peak) occurs when driving frequency equals the natural frequency
The amplitude increases with decreased damping
The curve broadens as the damping increases
The shape of the resonance curve depends on b
0
22 2
0
FmA
When the driving vibration matches
the natural frequency of an object, it
produces a Sympathetic Vibration -
it Resonates!
Natural Frequency & Resonance
http://www.youtube.com/watch?v=17tqXgvCN0E
A singer or musical instrument can shatter a crystal
goblet by matching the goblet’s natural oscillation
frequency.
Natural Frequency & Resonance
All objects have a natural frequency of
vibration or oscillation. Bells, tuning
forks, bridges, swings and atoms all
have a natural frequency that is related
to their size, shape and composition.
A system being driven at its natural
frequency will resonate and produce
maximum amplitude and energy.
https://www.youtube.com/watch?v=KqqyAZDpV6c
© 2013 Pearson Education, Inc.
QuickCheck 14.15
A. The red oscillator.
B. The blue oscillator.
C. The green oscillator.
D. They all oscillate for
the same length of time.
The graph shows how three oscillators respond as the frequency
of a driving force is varied. If each oscillator is started and then
left alone, which will oscillate for the longest time?
© 2013 Pearson Education, Inc.
QuickCheck 14.15
A. The red oscillator.
B. The blue oscillator.
C. The green oscillator.
D. They all oscillate for
the same length of time.
The graph shows how three oscillators respond as the frequency
of a driving force is varied. If each oscillator is started and then
left alone, which will oscillate for the longest time?
© 2013 Pearson Education, Inc.
Driven Oscillations and Resonance
Consider an oscillating system that, when left to itself,
oscillates at a natural frequency f0.
Suppose that this system is subjected to a periodic
external force of driving frequency fext.
The amplitude of oscillations
is generally not very high
if fext differs much from f0.
As fext gets closer and closer
to f0, the amplitude of the
oscillation rises dramatically.
A singer or musical instrument can shatter a crystal
goblet by matching the goblet’s natural oscillation
frequency.