Pendulum Lecturer: Professor Stephen T. Thornton

Pendulum

Lecturer: Professor Stephen T. Thornton

Reading Quiz

What is happening to the bridge in this photo?

A) A ship passing under the bridge has just hit it.

B) This is a fake photo for one of the thriller movies.

C) The wind is causing a forced resonant oscillation.

D) This is a painting, not a photo.

C)This is a photo of the Tacoma Narrows bridge collapse of the 1940s. We will watch a video of it today. The bridge oscillated in resonance and eventually broke apart.

Last Time

Oscillations

Simple harmonic motion

Periodic motion

Springs

Energy

Today

Simple pendulum

Physical pendulum

Damped and forced oscillations

Motion of a Pendulum

0U

(1 cos )U mgL

Small Angles

24

22

(1 cos )

Small angles: cos 1 ( )2

1 (1 )2 2

U mgL

O

mgLU mgL

This is a parabola. Pendulum has similar potential energy to a spring.

0

The Potential Energy of a Simple Pendulum

(1 cos )U mgL

The Simple Pendulum

Position of mass along arc:

Velocity along the arc:

Tangential acceleration:

s

ds dvdt dt

22

dv dadt dt

L

The tangential restoring force comes from gravity (tension is always centripetal for a pendulum):

tan

tan sin

But = , and / , so we have

= mg

F mg mg

x xmgF x k

We have a restoring force F = -kx for small angle oscillations, which is like Hooke’s law, so we have simple harmonic motion!

max

1

Let's find the solution in terms of , not .

cos( )

where

so we have

12 21 2

Remember this is all true for small angles .

x

t tmg gk

m

gf

T gf

mg

//

xk mg

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The Simple Pendulum

2Tg

This is a remarkable result. The period only depends on the length of the pendulum, not the mass! Galileo figured this out as a young man sitting in church while watching the chandeliers swing.

Energy of a simple pendulum:

h2 2

2 22

22 2

( ) (1 cos )

1 1( ) 1 (1 ...)2 2

1 1 1( )2 2 2

1 1( ) and ( )2 2

U mgh mg

U mg mg

dx dK mv m mdt dt

dK m U mgdt

x

http://physics.bu.edu/~duffy/semester1/semester1.html

Conceptual Quiz:A person sits on a playground swing. When pushed gently once, the swing oscillates back and forth at its natural frequency. If, instead, two people sit side by side on the swing, the new natural frequency of the swing is 

A) greater.B) smaller.C) the same.

Answer: C

The problem statement indicated it is a gentle push, so we assume small oscillations. In that case, the period doesn’t depend on the mass, only the length of the swing.

2Tg

Conceptual Quiz:Grandfather clocks have a weight at the bottom of the pendulum arm that can be moved up or down to correct the time. Suppose that your grandfather clock runs slow. In which direction do you move the weight to correct the time on the clock?

A) upB) downC) moving the weight does not matter.D) throw the clock away and get a new one, because physics is too hard.

2Tg

A) up In order for the clock to run faster, we

want the time between ticks to be smaller. That is, we want the period to decrease. In order to do that we decrease L which decreases the period. We adjust a small screw usually on the bottom of the pendulum arm that raises the weight (mass bob). This decreases L and makes the clock run faster.

Simple Pendulum. What is the period of a simple pendulum 53 cm long (a) on the Earth, and (b) when it is in a freely falling elevator?

Examples of Physical Pendulums

Demo

A physical pendulum is any real extended object that oscillates back and forth.

The torque about point O is:

Substituting into Newton’s second law for rotation gives:

The Physical Pendulum

sinmght q=-

2

2sin

dI mgh

dt

qq=-

For small angles, this becomes:

which is the equation for SHM, with

max cos( )

2

mghI

t

ITmgh

2

20

d mgh

dt I

qq

æ ö÷ç+ =÷ç ÷çè ø

Conceptual Quiz:A simple pendulum oscillates with a maximum angle to the vertical of 5o. If the same pendulum is repositioned so that its maximum angle is 7o, we can say that

A) both the period and the energy are unchanged.B) both the period and the energy increase.C) the period is unchanged and the energy increases.D) the period increases and the energy is unchanged.E) none of these is correct.

Answer: C

This is a small oscillation, and for small oscillations, the period does not change significantly.

The weight moves further up in elevation, and its U increases, so its total energy also increases.

Conceptual QuizConceptual Quiz

A hole is drilled through the

center of Earth and emerges

on the other side. You jump

into the hole. What happens

to you ?

A) you fall to the center and stop

B) you go all the way through and continue off into space

C) you fall to the other side of Earth and then return

D) you won’t fall at all

You fall through the hole. When you reach the

center, you keep going because of your inertia.

When you reach the other side, gravity pulls gravity pulls

you back toward the centeryou back toward the center. This is Simple This is Simple

Harmonic Motion!Harmonic Motion!

Conceptual QuizConceptual Quiz

A hole is drilled through the

center of Earth and emerges

on the other side. You jump

into the hole. What happens

to you ?

A) you fall to the center and stop

B) you go all the way through and continue off into space

C) you fall to the other side of Earth and then return

D) you won’t fall at all

Follow-up:Follow-up: Where is your acceleration zero? Where is your acceleration zero?

A mass oscillates in simple

harmonic motion with amplitude

A. If the mass is doubled, but the

amplitude is not changed, what

will happen to the total energy of

the system?

A) total energy will increase

B) total energy will not change

C) total energy will decrease

Conceptual QuizConceptual Quiz

A mass oscillates in simple

harmonic motion with amplitude

A. If the mass is doubled, but the

amplitude is not changed, what

will happen to the total energy of

the system?

A) total energy will increase

B) total energy will not change

V) total energy will decrease

The total energy is equal to the initial value of the

elastic potential energy, which is PEs = kA2. This

does not depend on mass, so a change in mass will not affect the energy of the system.

Conceptual QuizConceptual Quiz

Follow-up:Follow-up: What happens if you double the amplitude? What happens if you double the amplitude?

12

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Damped harmonic motion is harmonic motion with a frictional or drag force. If the damping is small, we can treat it as an “envelope” that modifies the undamped oscillation.

Damped Harmonic Motion

dampingIf ,

then is Newton's 2nd law

F bv

ma kx bv

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This gives

If b is small, a solution of the form

will work, with

Damped Harmonic Motion

2

20

d x dxm b kx

dt dt+ + =

cos 'tx Ae tg w-=

2

2

2

'4

b

m

k b

m m

g

w

=

= -

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If b2 > 4mk, ω’ becomes imaginary, and the system is overdamped (C).

For b2 = 4mk, the system is critically damped (B) —this is the case in which the system reaches equilibrium in the shortest time.

Case A (b2 < 4mk) is underdamped; it oscillates within the exponential envelope.

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There are systems in which damping is unwanted, such as clocks and watches.

Then there are systems in which it is wanted, and often needs to be as close to critical damping as possible, such as automobile shock absorbers, storm door closures, and earthquake protection for buildings..

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Forced vibrations occur when there is a periodic driving force. This force may or may not have the same period as the natural frequency of the system.

If the frequency is the same as the natural frequency, the amplitude can become quite large. This is called resonance.

Forced Oscillations; Resonance

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The equation of motion for a forced oscillator is:

The solution is:

where

and

0 cosma kx bv F tw=- - +

0 0sin( )x A tw f= +

( )0

0 22 2 2 2 20 /

FA

m b mw w w=

- +

( )

2 21 0

0 tan/b m

w wf

w-æ ö- ÷ç ÷= ç ÷ç ÷÷çè ø

Show hacksaw blade resonance demo. (Go back and show previous slide.)

Do damping and forced oscillation demo. (Go back and show previous slide.)

Show Tacoma Narrows Bridge collapse.

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The sharpness of the resonant peak depends on the damping. If the damping is small (A) it can be quite sharp; if the damping is larger (B) it is less sharp.

Like damping, resonance can be wanted or unwanted. Musical instruments and TV/radio receivers depend on it.

Human Leg. The human leg can be compared to a physical pendulum, with a “natural” swinging period at which walking is easiest. Consider the leg as two rods joined rigidly together at the knee; the axis for the leg is the hip joint. The length of each rod is about the same, 55 cm. The upper rod has a mass of 7.0 kg and the lower rod has a mass of 4.0 kg. (a) Calculate the natural swinging period of the system. (b) Check your answer by standing on a chair and measuring the time for one or more complete back-and-forth swings. The effect of a shorter leg is a shorter swinging period, enabling a faster “natural” stride.

Unbalanced Tires. An 1150 kg automobile has springs with k = 16,000 N/m. One of the tires is not properly balanced; it has a little extra mass on one side compared to the other, causing the car to shake at certain speeds. If the tire radius is 42 cm, at what speed will the wheel shake most?