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## Pendulum

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**A pendulum pivots at the top of the string.**The forces on a pendulum are due to gravity and tension. Tension exerts no torque Gravity exerts a torque Force to Torque q L FT mgsin q mg**The moment of inertia for a single mass is I = mr2.**The angular acceleration is due to the torque. Compare angle and sine Angle(rad) Sine 1 (0.01745) 0.01745 2 (0.03491) 0.03490 5 (0.08727) 0.08716 10 (0.17453) 0.17365 15 (0.26180) 0.25882 20 (0.34907) 0.34202 30 (0.52360) 0.50000 For small angles sinq = q. Small Angles**Simple Pendulum**• Angular acceleration and angle are related as a simple harmonic oscillator. • k = mg/L • The angular frequency and period are q L m**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/s2 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. Tarzan L = 25 m 2A = 8 m**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. Vine Tension L v2/r FT A mg**Real pendulums have mass over the whole length.**Use the actual moment of inertia Physical Pendulum**Damped Harmonic Motion**• Real oscillators lose amplitude each cycle. • Friction force exists • This is called damping • Underdamped oscillator • Greater damping can eliminate all oscillations. • Overdamped oscillator • Critical damping (A)**Energy Damping**• Real pendulums have friction. • The damping is due to the work from friction. • Measure energy loss at maximum amplitude**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 period • A little force can get a large amplitude