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.