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- 1. Lightening Review Torque & Static Equilibrium
- 2. Which mass is heavier? 1. The hammer portion. 2. The handle portion. 3. They have the same mass. Balance Point Cut at balance point
- 3. Locate the Center of Mass 1. 4 Meters 2. 5 Meters 3. 8 Meters M X=0 X=10M 10N 40N
- 4. Periodic Motion Chapter 13
- 5. Periodic systems are all around us • Rising of the sun • Change of the seasons • The tides • Bird songs • Rotation of a bicycle wheel Periodic motion is any motion that repeats on a regular time basis. Time Position Period, T
- 6. Periodic Motion t, time Amplitude Period, T The motion PERIOD is the time, T, to return to same point. The FREQUENCY, f, is the inverse of the period, 1/T. The AMPLITUDE, A, is the maximum displacement. A Since the motion returns to the same point at t=T, it must be true that 2 T so 2 1 T f
- 7. Harmonic Motion: Key concepts • Harmonic motion is an important and common type of repetitive, or “oscillatory” motion • Harmonic motion is “sinusoidal” • Oscillatory motion results when an applied force (1) depends on position AND (2) reverses direction at some position. • The two most common harmonic motions are the pendulum and the spring-mass system.
- 8. A history of time (keeping)
- 9. Modern clocks Mechanical, quartz, and atomic.
- 10. Clocks use Harmonic Motion • Stonehenge, a clock based on the sun • Sundials • Water clocks • Pendulum based clocks • Geneva escapement mechanism • Quartz crystal clocks • Atomic clocks Demonstrate some “clocks”
- 11. Our journey begins with uniform circular motion. • Uniform Circular Motion is closely related to Harmonic Motion (oscillations) • Objects in UCM have a constant centripetal acceleration (ac=V2/R). Dq 1 V 2 V R R Dq V D V R V a R V V t V V t V t V V V V R V c 2 sin D D D D D D D D D q q q
- 12. Theme-park Physics • What is the angular speed and linear speed needed to have a rider feel “zero G” at the top of the ride? • What is the net acceleration at the bottom of the ride? R TOP: BOTTOM: ac ac ac=g=V2/R=R2 NOTE: There is more to this than meets the eye! The force of the ride on the rider is zero at the top of the ride, and is Mg at the bottom of the ride. F=Mac=Mg Mg Mg Fc+Fg=2Mg If the ride is 9.8 meters in radius, =sqrt(g/R)=1 rad/sec V=9.8 m/sec (about 20 MPH)
- 13. Theme-park Physics: Feeling weightless • What is the angular speed and linear speed needed to have a rider feel “zero G” at the top of the ride? • What is the net acceleration at the bottom of the ride? R ac g g=ac ac=g=V2/R=R2 Vzero-G = sqrt(Rg) We don’t feel the acceleration of gravity acting on our bodies, only the force of gravity of the floor pushing up against gravity. Weightlessness is “zero g” acceleration, normal gravity is “one g”. Unconsciousness can result at around 9 g without special equipment.
- 14. Connection of rotation and harmonic motion • Physlet Illustration 16.1
- 15. Connection between rotational and oscillatory motion Y motion X motion q q sin cos R y R x q t R y t R x sin cos t t
- 16. Harmonic motion, velocity q V Vx V q t R V Vx q sin sin R V So, now have… t R V t R x x sin cos
- 17. Harmonic motion, acceleration q ac ac ax q t R t R V a ax q cos cos cos 2 2
- 18. Harmonic motion-summary t R a t R v t R x x x cos sin cos 2 Look at what this says…. x ax 2 Or x m ma F x x 2 So, we have a force that depends on position, and reverses direction at x=0.
- 19. Spring-time remembered…. We know that for an ideal spring, the force is related to the displacement by kx F But we just showed that harmonic motion has x m F 2 So, we directly find out that the “angular frequency of motion” of a mass-spring system is m k m k 2
- 20. Harmonic motion: all together now. t A a t A v t A x x x cos sin cos 2 t, time x v a
- 21. Mass on a spring: x, v and a.
- 22. Properties of Mass-Spring System • Physlet Exploration 16-1. • How does the period of oscillation change with amplitude? Exploration 16-1
- 23. Application: Tuning Forks and Musical Instruments • A tuning fork is basically a type of spring. The same is true for the bars that make up a xylophone. They have a very large spring constant. • Since the oscillation frequency does not change with amplitude, the tone of the tuning fork and xylophone note is independent of loudness.
- 24. Simulation: Mass on Spring • Physlet Illustration 16-4: forced & damped motion of spring/mass system. Physlet Illustration 16-4
- 25. The pendulum: keeping time harmonically. Mg T q Ftangent q q Mg Mg Ft sin L s q
- 26. Small angle approximation 0 sin q q q Theta must be calculated in RADIANS! Generally the approximation is used for angles less than 30 degrees, or about ½ radian.
- 27. Compare pendulum and spring. Pendulum Spring L s Mg F Kx F Force Forces depend on position, reverse direction at some position. Periodic motion: t x x cos 0 t s s sin 0 Angular frequency: M K or t q q sin 0 L g