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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.

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Presentation on theme: "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."— Presentation transcript:

1 Pendulum

2 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

3 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  = .

4 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

5 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

6 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

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

8 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

9 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


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