Physics 121 for Majors Class 18 Linear Harmonic Last Class We saw how motion in a circle is mathematically similar to motion in a straight line. We learned that there is a centripetal acceleration (and force) and a tangential acceleration (and force) when objects move in circular paths. We learned that the magnitude of the centripetal force - whatever the real force is that keeps an object moving in a circular path -is / We also learned how to deal with motion as viewed from an accelerating reference frame. Today s Class Any equation of the form gives rise to simple harmonic (sine and cosine) motion. Know the solutions! Springs and pendulums are typical. Vertical springs oscillate essentially like horizontal springs. Energy oscillates between kinetic and potential energies. We ll see how otation and oscillation are similar mathematically. Damped oscillators: overdamping, underdamping, critical damping Forced oscillators: resonance Section 1 Solving the Spring Problem The Spring Force The spring can be compressed or stretched. The force is always opposite the displacement. If 0 is the equilibrium position 1
The Spring Force If 0 is some other position where is the equilibrium position The Equation of Motion Given the force, we can find the equation that governs : The Equation of Motion When you take two derivatives of with respect to time you get minus a constant times the same function. What is the function? The cos or we can write the same thing as cossin Where,,, are constants to be determined by the initial conditions. The cos We stretch a spring a distance and release it from rest. What are and? 0 0 0 First, find the velocity: sin 0 0 sin 0 0 cos0 cos The cos Let s check the solution: 0 sin 0 0 cos 2
Section 2 Interpreting the The General How do we understand the solution? cos The General How do we understand the solution? cos is the amplitude The General If this is what is 3? The General How do we understand the solution? cos is the amplitude is the phase angle it determines how we shift the sine wave right or left. Phase Angle 3 cos 4 3cos4/2 3
Phase Angle 3 cos 4 3cos4/2 The General The period of the function (seconds per cycle) is the time between two maxima: 2 4 2 The General The frequency of the function (cycles per second) is the reciprocal of the period: 2 The General In one cycle there are 2 radians. That means that is the angular frequency. It tells how many radians of a sine function the system oscillates in one second but there is no rotational motion, of course. 2 2 2 The General How do we understand the solution? cos is the amplitude is the phase angle it determines how we shift the sine wave right or left. is the angular frequency It is useful mathematically It is related to frequency and period by 2 2/ Section 3 Vertical Spring 4
No Mass Assume the spring is massless. We call the bottom of the spring 0. Static Equilibrium When we add a mass, the spring stretches. The gravitational and spring forces sum to zero: 0 Net Force Pull the spring down a distance. 0and the additional force is up, so: Net Force Pull the spring up a distance. 0and the additional force is down. Net Force If we take Y to be the displacement from equilibrium, the motion is the same as horizontal spring: Section 4 Energy cos 5
Potential Energy We ve already seen this: 1 2 We choose the constant of integration to be zero, so 1 2 1 2 cos Kinetic Energy 1 2 1 2 sin Since, we can also write this as 1 2 sin Total Energy We just add these together: 1 2 cos 1 2 sin 1 2 Note that the energy is not a function of time. Energy Flow Total energy is a constant flowing back and forth between kinetic and potential energies. We can see this in the potential energy well. This has k =12 N/m. U E K Section 5 Rotation and A Rotating Object Consider a conical pendulum with a vector pointing from the center of the circle to the mass. 6
A Rotating Object Consider the y- component of the rotating vector. Graphing the y-component Now graph the y-component of the vector as a function of time. The y-component moves like a simple harmonic oscillator. Section 6 Linearization Linear Force Whenever a force as the form, we say it is a linear force and the system is a linear harmonic oscillator. There are many forces in nature like that. Linear Regimes Even when forces aren t linear, they may be close to linear for small amplitude oscillations. Consider the potential energy of an atom in a crystal lattice: U r Linear Regimes Near the bottom of the well, the force is linear and the potential energy is a parabola U r 7
Taylor Series The way we usually linearize functions is to use a Taylor Series. 0 1 1! 0 1 2! 0 1 3! 0 Taylor Series for sin(x) 0 1 1! 0 1 2! 0 1 3! 0 sin sin 0 1 1 cos 0 1 2 sin 0 1 cos 0 6 sin 1 6 This only works if the derivatives are well behaved and 1. It works best if 0. Or if we only keep the linear term sin But this is only good for small values of. Section 7 Pendulum The Pendulum Equation After we ve talked about torques, we ll find that a pendulum satisfies the equation: sin Note that this isn t a linear equation, but we can linearize it L θ The Pendulum Equation sin L θ The Pendulum Equation cos L θ What is the solution? If we move the pendulum to and release it from rest: cos 8
Section 8 Damped and Driven Real The oscillators we have encountered so far have no energy losses. If we start the system oscillating with an amplitude A, it continues to oscillate with that amplitude forever. To include damping in the oscillator equation, we include a term that is dependent on velocity, much like drag. Damped Damped oscillators have three kinds of behavior depending on the size of the damping coefficient: Underdamped Critically damped Overdamped Damped s Let If If If underdamped critically damped overdamped 2 Underdamping cos There are oscillations that die out exponentially Overdamping There is no overshooting (top curve) 9
Critical Damping This is the fastest damping without overshooting (middle curve) Driven Oscillator sin Initially the solution is a complicated combination of oscillations at the natural frequency and the driving frequency. Eventually, the system oscillates at the driving frequency: cos Driven Oscillator The amplitude depends on the driving force, the natural frequency, and the damping constant: 0 Driven Oscillator 0 Section 9 Recap Big Ideas s of are cosfind the constants, understand all details of the motion! Know spring and pendulum examples. Rotation and oscillation are similar mathematically. Know how energy oscillates between kinetic and potential energies. Understand overdamping, underdamping, and critical damping Understand resonance 10
Schedule Do Post-Class Quiz #18 Do Pre-Class Quiz #19 HW #17 is due Friday HW #18 is due Wednesday Quiz #5 is due Saturday 11