LAST TIME: Simple Pendulum: The displacement from equilibrium, x is the arclength s = L. s / L x / L Accelerating & Restoring Force in the tangential direction, taking cw as positive initial displacement direction and assume small angle approximation: F mg sin mg x F mg ma L g a x L 2 x g L 2 L g
STRATEGY: If you can show that the system obeys Hooke s Law: Force ~ - Displacement Then you get to ASSUME the system moves in SHM and that: 2 OR a 2 x So you simplify the equation down to this and whatever is the coefficient of x is the square of ω, the angular frequency!!!
Physical Pendulum: Rods & Disks If a hanging object oscillates about a fixed axis that does not pass through the center of mass and the object cannot be approximated as a particle, the system is called a physical pendulum It cannot be treated as a simple pendulum The gravitational force provides a torque about an axis through O The magnitude of the torque is mgd sin I is the moment of inertia about the axis through O
Physical Pendulm Sample Problem 1. A uniform thin rod (length L = 1.0 m, mass = 2.0 kg) is suspended from a pivot at one end. Assuming small oscillations, derive an expression for the angular frequency in terms of the given variables (m, L, g), and then solve for a numerical value in rad/s. Show all your work. Sketch a diagram showing angle, lengths, lever arms, etc, and explain whatever is needed for a fantastic solution.
The Physical Pendulum Any solid object that swings back and forth under the influence of gravity can be modeled as a physical pendulum. The gravitational torque for small angles ( 10) is: Plugging this into Newton s second law for rotational motion, I, we find the equation for SHM, with: 2013 Pearson Education, Inc. Slide 14-82
Physical Pendulm Sample Problem 2. A uniform disk (R = 1.0 m, m = 2.0 kg) is suspended from a pivot a distance 0.25 m above its center of mass. Ignore air resistance and any other frictional forces. Starting from Newton s Second Law and assuming small oscillations, derive a reduced expression for the angular frequency in terms of the given variables: (R, m, g), and then solve for a numerical value in rad/s. Show all your work. Sketch a diagram showing angle, lengths, lever arms, etc, and explain whatever is needed for a fantastic solution.
QuickCheck 14.14 A solid disk and a circular hoop have the same radius and the same mass. Each can swing back and forth as a pendulum from a pivot at one edge. Which has the larger period of oscillation? A. The solid disk. B. The circular hoop. C. Both have the same period. D. There s not enough information to tell. Slide 14-85
QuickCheck 14.14 A solid disk and a circular hoop have the same radius and the same mass. Each can swing back and forth as a pendulum from a pivot at one edge. Which has the larger period of oscillation? A. The solid disk. B. The circular hoop. C. Both have the same period. D. There s not enough information to tell. Slide 14-86
A hoop made of a thin wire of mass M and radius R is pinned at its edge as shown. Find the period of oscillation.
Damped Oscillations Position-versus-time graph for a damped oscillator. 2013 Pearson Education, Inc. Slide 14-89
Driven Oscillations and Resonance A F 0 2 2 0 m 2 b m 2 F 0 is the driving force 0 is the natural frequency of the undamped oscillator b is the damping constant The figure shows the same oscillator with three different values of the damping constant. The resonance amplitude becomes higher and narrower as the damping constant decreases. 2013 Pearson Education, Inc.
Resonance Resonance (maximum peak) occurs when driving frequency equals the natural frequency The amplitude increases with decreased damping The curve broadens as the damping increases The shape of the resonance curve depends on b A F 0 m 2 2 0 2
Natural Frequency & Resonance When the driving vibration matches the natural frequency of an object, it produces a Sympathetic Vibration - it Resonates! A singer or musical instrument can shatter a crystal goblet by matching the goblet s natural oscillation frequency. http://www.youtube.com/watch?v=17tqxgvcn0e
Natural Frequency & Resonance All objects have a natural frequency of vibration or oscillation. Bells, tuning forks, bridges, swings and atoms all have a natural frequency that is related to their size, shape and composition. A system being driven at its natural frequency will resonate and produce maximum amplitude and energy. https://www.youtube.com/watch?v=kqqyazdpv6c
QuickCheck 14.15 The graph shows how three oscillators respond as the frequency of a driving force is varied. If each oscillator is started and then left alone, which will oscillate for the longest time? A. The red oscillator. B. The blue oscillator. C. The green oscillator. D. They all oscillate for the same length of time. 2013 Pearson Education, Inc.
QuickCheck 14.15 The graph shows how three oscillators respond as the frequency of a driving force is varied. If each oscillator is started and then left alone, which will oscillate for the longest time? A. The red oscillator. B. The blue oscillator. C. The green oscillator. D. They all oscillate for the same length of time. 2013 Pearson Education, Inc.
Driven Oscillations and Resonance Consider an oscillating system that, when left to itself, oscillates at a natural frequency f 0. Suppose that this system is subjected to a periodic external force of driving frequency f ext. The amplitude of oscillations is generally not very high if f ext differs much from f 0. As f ext gets closer and closer to f 0, the amplitude of the oscillation rises dramatically. 2013 Pearson Education, Inc. A singer or musical instrument can shatter a crystal goblet by matching the goblet s natural oscillation frequency.