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Transcript
Lecture 17 Oscillations Today’s Topics: • • • • Periodic motion (Simple Harmonic Motion) Springs and pendulums Energy Damped and driven motion Restoring Forces No restoring force Restoring force returns the ball to equilibrium ACT: to the center of Earth A hole is drilled through the a) you fall to the center and stop center of Earth and emerges b) you go all the way through and continue off into space on the other side. You jump into the hole. What happens to you ? c) you fall to the other side of Earth and then return d) you won’t fall at all You fall through the hole. When you reach the center, you keep going because of your inertia. When you reach the other side, gravity pulls you back toward the center. Gravity is a restoring force here! Follow-up: Where is your acceleration zero? How do we describe these oscillations about equilibrium? x = A cos q But q = wt , so x = A cos wt Simple Harmonic Motion x = A cos wt 2p w = 2p f = T 1 f = T amplitude A: the maximum displacement period T: the time required to complete one cycle frequency f: the number of cycles per second (measured in Hz) ACT: simple harmonic motion A mass on a spring in SHM has amplitude A and period T. What is the net displacement of the mass after a time interval T? a) 0 b) A/2 c) A d) 2A e) 4A The displacement is Δx = x2 – x1. Because the initial and final positions of the mass are the same (it ends up back at its original position), then the displacement is zero. Example The position of a simple harmonic oscillator is given by x(t ) = (0.50 m) cos (p3 t ) where t is in seconds. What is the period of the oscillator? x = A cos wt A = 0.5 m w= p 3 rad/s Velocity and Acceleration Where is vmax? ax = - ! Aw cos wt 2 amax Where is amax? Springs HOOKE’S LAW: RESTORING FORCE OF AN IDEAL SPRING The restoring force on an ideal spring is Fx = - k x How do we determine ω for a spring? The frequency is determined by the physical properties of the system x = A cos w t a x = - Aw 2 cos w t å F = -kx = ma - kA = -mAw k w= m 2 x ACT: spring on the Moon A mass oscillates on a vertical spring with period T. If the whole setup is taken to the Moon, how does the period change? a) period will increase b) period will not change c) period will decrease The period of simple harmonic motion depends only on the mass and the spring constant and does not depend on the acceleration due to gravity. By going to the Moon, the value of g has been reduced, but that does not affect the period of the oscillating mass–spring system. To measure the mass of an astronaut on the space station they employ a device that consists of a spring-mounted chair in which the astronaut sits. The spring has a spring constant of 606 N/m and the mass of the chair is 12.0 kg. The measured period is 2.41 s. Find the mass of the astronaut. k w= mtotal mtotal = k w 2 mchair + mastro k = (2p T )2 mastro k = - mchair 2 (2p T ) ( 606 N m )(2.41 s ) = 2 4p 2 - 12.0 kg = 77.2 kg Springs and Energy DEFINITION OF ELASTIC POTENTIAL ENERGY The elastic potential energy is the energy that a spring has by virtue of being stretched or compressed. For an ideal spring, the elastic potential energy is PE elastic = 12 kDx 2 SI Unit of Elastic Potential Energy: joule (J) Total Mechanical Energy 1 2 1 E = mv + kDx 2 2 2 As a function of time, The total energy is constant; as the kinetic energy increases, the potential energy decreases, and vice versa. Since we know the position and velocity as functions of time, we can find the maximum kinetic and potential energies: The Pendulum The restoring force of a pendulum is proportional to sin θ, whereas the restoring force for a spring is proportional to the displacement (which is θ in this case). Frestoring = −mgsin θ = ma However, for small angles, sinθ and θ are approximately equal (small angle approximation) Substituting θ for sin θ allows us to treat the pendulum in a mathematically identical way to the mass on a spring. We find that the period of a pendulum depends on the length of the string and g: ACT: period of a pendulum Two pendulums have the same length, but different masses attached to the string. How do their periods compare? a) period is greater for the greater mass b) period is the same for both cases c) period is greater for the smaller mass The period of a pendulum depends on the length and the acceleration due to gravity, but it does not depend on the mass of the bob. L = 2π T g Follow-up: What happens if the amplitude is doubled? ACT: pendulum on the moon A swinging pendulum has period T on Earth. If the same pendulum a) period increases were moved to the Moon, how b) period does not change does the new period compare to c) period decreases the old period? The acceleration due to gravity is smaller on the Moon. The relationship between the period and g is given by: T = 2π therefore, if g gets smaller, T will increase. L g ACT: grandfather clock A grandfather clock has a weight at the bottom of the pendulum that can be moved up or down. If the clock is running slow, what should you do to adjust the time properly? a) move the weight up b) move the weight down c) moving the weight will not matter d) call the repairman The period of the grandfather clock is too long, so we need to decrease the period (increase the frequency). To do this, the length must be decreased, so the adjustable weight should be moved up in order to shorten the pendulum length. L = T 2π g Damped Harmonic Motion 1) simple harmonic motion 2&3) underdamped 4) critically damped 5) overdamped Driven Harmonic Motion and Resonance When a force is applied to an oscillating system at all times, the result is driven harmonic motion. Here, the driving force has the same frequency as the spring system and always points in the direction of the object’s velocity. Resonance occurs when the frequency of the force matches a natural frequency at which the object will oscillate.