Pendulum

Force to Torque  A pendulum pivots at the top of the string.  The forces on a pendulum are due to gravity and tension. Tension exerts no torqueTension exerts no torque Gravity exerts a torqueGravity exerts a torque mg FTFT mg sin   L

Small Angles  The moment of inertia for a single mass is I = mr 2.  The angular acceleration is due to the torque.  Compare angle and sine Angle(rad)Sine 1  ( )  ( )  ( )  ( )  ( )  ( )  ( )  For small angles sin  = .

Simple Pendulum  Angular acceleration and angle are related as a simple harmonic oscillator. k = mg/Lk = mg/L  The angular frequency and period are m  L

Tarzan  Tarzan is going to swing from one branch to another 8 m away at the same height using a vine which is 25 m long.  How long does the swing take?  Tarzan forms a pendulum and the period will be  Using 25 m and 9.8 m/s 2 T = 10. s  The other branch is half a period, t = 5.0 s.  Note that the mass or distance to the branch didn’t affect the time. L = 25 m 2A = 8 m

Vine Tension  What is the maximum tension of the vine in the previous problem?  The maximum occurs at the bottom with maximum centripetal acceleration.  Find the tension using circular motion. L FTFT v 2 /r mg A

Physical Pendulum  Real pendulums have mass over the whole length. Use the actual moment of inertiaUse the actual moment of inertia

Damped Harmonic Motion  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 dampingThis is called damping

Resonance  Work can also be done to increase the energy.  If it’s synchronized to the natural frequency then the 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 next