PendulumPendulum

Force to TorqueForce to Torque

A pendulum pivots at the top A pendulum pivots at the top of the string.of the string.

The forces on a pendulum The forces on a pendulum are due to gravity and are due to gravity and tension.tension.• Tension exerts no torqueTension exerts no torque

• Gravity exerts a torqueGravity exerts a torque

mg

FT

mg sin

L

sinLmg

Small AnglesSmall Angles

The moment of inertia for a The moment of inertia for a single mass is single mass is II = = mrmr22..

The angular acceleration is The angular acceleration is due to the torque.due to the torque.

Compare angle and sineCompare angle and sine• Angle(rad)Angle(rad)

SineSine

• 11 (0.01745) (0.01745) 0.017450.01745

• 22 (0.03491) (0.03491) 0.034900.03490

• 55 (0.08727) (0.08727) 0.087160.08716

• 1010 (0.17453) (0.17453) 0.173650.17365

• 1515 (0.26180) (0.26180) 0.258820.25882

• 2020 (0.34907) (0.34907) 0.342020.34202

• 3030 (0.52360) (0.52360) 0.500000.50000 For small angles sinFor small angles sin= = ..

L

mgm

LmgmL

LmgI

sin

sin2

Simple PendulumSimple Pendulum

Angular acceleration and Angular acceleration and angle are related as a simple angle are related as a simple harmonic oscillator.harmonic oscillator.• kk = = mgmg//LL

The angular frequency and The angular frequency and period are period are m

L

L

mgm gLT

Lgmk

/2/2

//

TarzanTarzan

Tarzan is going to swing Tarzan is going to swing from one branch to another 8 from one branch to another 8 m away at the same height m away at the same height using a vine which is 25 m using a vine which is 25 m long.long.

How long does the swing How long does the swing take?take?

Tarzan forms a pendulum Tarzan forms a pendulum and the period will be and the period will be

Using 25 m and 9.8 m/sUsing 25 m and 9.8 m/s22

• TT = 10. s = 10. s The other branch is half a The other branch is half a

period, period, tt = = 5.0 s5.0 s.. Note that the mass or Note that the mass or

distance to the branch didn’t distance to the branch didn’t affect the time. affect the time.

gLT /2

L = 25 m

2A = 8 m

Vine TensionVine Tension

What is the maximum What is the maximum tension of the vine in the tension of the vine in the previous problem?previous problem?

The maximum occurs at the The maximum occurs at the bottom with maximum bottom with maximum centripetal acceleration.centripetal acceleration.

Find the tension using Find the tension using circular motion.circular motion.

)/1(

)(

)(

22

2

2

2

22

22

LAmgF

L

gmAmgFF

L

gmA

L

gA

L

mma

L

A

L

va

T

T

L

FT

v2/r

mgA

Physical PendulumPhysical Pendulum

Real pendulums have mass Real pendulums have mass over the whole length.over the whole length.• Use the actual moment of Use the actual moment of

inertiainertia

I

mgL

LmgI

sin

Damped Harmonic MotionDamped Harmonic Motion

Real pendulums lose amplitude with each swing.Real pendulums lose amplitude with each swing.• Friction force existsFriction force exists• Measure energy loss at maximum amplitudeMeasure energy loss at maximum amplitude• This is called This is called dampingdamping

frictionWL

mgxE

L

mgxkxE

2

22

22

21

0

ResonanceResonance

Work can also be done to increase the energy.Work can also be done to increase the energy.

If it’s synchronized to the natural frequency then the If it’s synchronized to the natural frequency then the system is in resonance. system is in resonance. • Pushing a swing at each periodPushing a swing at each period• A little force can get a large amplitudeA little force can get a large amplitude

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