SIMPLE HARMONIC MOTION: NEWTON’S LAW PH421:Oscillations F09 4/17/2017 SIMPLE HARMONIC MOTION: NEWTON’S LAW simple not simple PRIOR READING: Main 1.1, 2.1 Taylor 5.1, 5.2 http://www.myoops.org/twocw/mit/NR/rdonlyres/Physics/8-012Fall-2005/7CCE46AC-405D-4652-A724-64F831E70388/0/chp_physi_pndulm.jpg Please do not distribute

The simple pendulum Energy approach q T m mg PH421:Oscillations F09 4/17/2017 The simple pendulum Energy approach q mg m T Please do not distribute

The simple pendulum Energy approach q T m mg PH421:Oscillations F09 4/17/2017 The simple pendulum Energy approach q mg m T Please do not distribute

The simple pendulum q T m Newton Known torque mg PH421:Oscillations F09 4/17/2017 The simple pendulum q mg m T Newton Known torque This is NOT a restoring force proportional to displacement (Hooke’s law motion) in general, but IF we consider small motion, IT IS! Expand the sin series … Please do not distribute

The simple pendulum in the limit of small angular displacements L q T PH421:Oscillations F09 4/17/2017 q mg m L T The simple pendulum in the limit of small angular displacements What is (t) such that the above equation is obeyed?  is a variable that describes position t is a parameter that describes time "dot" and "double dot" mean differentiate w.r.t. time g, L are known constants, determined by the system. Please do not distribute

C, p are unknown (for now) constants, possibly complex PH421:Oscillations F09 REVIEW PENDULUM 4/17/2017 q mg m L T C, p are unknown (for now) constants, possibly complex Substitute: p is now known (but C is not!). Note that w0 is NOT a new quantity! It is just a rewriting of old ones - partly shorthand, but also "w" means "frequency" to physicists! Please do not distribute

Simple harmonic motion PH421:Oscillations F09 REVIEW PENDULUM 4/17/2017 q mg m L T TWO possibilities …. general solution is the sum of the two and it must be real (all angles are real). If we force C' = C* (complex conjugate of C), then x(t) is real, and there are only 2 constants, Re[C], and Im[C]. A second order DEQ can determine only 2 arbitrary constants. Simple harmonic motion Please do not distribute

PH421:Oscillations F09 REVIEW PENDULUM 4/17/2017 q mg m L T Re[C], Im[C] chosen to fit initial conditions. Example: q(0) = 0 rad and dqdt(0) = 0.2 rad/sec Please do not distribute

REVIEW PENDULUM L q T m mg PH421:Oscillations F09 4/17/2017 Please do not distribute

PH421:Oscillations F09 4/17/2017 Remember, all these are equivalent forms. All of them have a known w0=(g/L)1/2, and all have 2 more undetermined constants that we find … how? Do you remember how the A, B, C, D constants are related? If not, go back and review until it becomes second nature! Please do not distribute

The simple pendulum Period does not depend on qmax, f L PH421:Oscillations F09 4/17/2017 q mg m L T The simple pendulum ("simple" here means a point mass; your lab deals with a plane pendulum) simple harmonic motion ( potential confusion!! A “simple” pendulum does not always execute “simple harmonic motion”; it does so only in the limit of small amplitude.) Period does not depend on qmax, f Please do not distribute

PH421: Oscillations; do not distribute Free, undamped oscillators – other examples 11/12/07 x m k L No friction I C q q mg m T Common notation for all Lecture 5/6 - damped oscllations

PH421:Oscillations F09 4/17/2017 • The following slides simply repeat the previous discussion, but now for a mass on a spring, and for a series LC circuit Please do not distribute

REVIEW MASS ON IDEAL SPRING PH421:Oscillations F09 REVIEW MASS ON IDEAL SPRING 4/17/2017 Newton x m k Particular type of force. m, k known Linear, 2nd order differential equation What is x(t) such that the above equation is obeyed? x is a variable that describes position t is a parameter that describes time "dot" and "double dot" mean differentiate w.r.t. time m, k are known constants Please do not distribute

REVIEW MASS ON IDEAL SPRING PH421:Oscillations F09 REVIEW MASS ON IDEAL SPRING 4/17/2017 x m k C, p are unknown (for now) constants, possibly complex Substitute: p is now known. Note that w0 is NOT a new quantity! It is just a rewriting of old ones - partly shorthand, but also “w” means “frequency” to physicists! Please do not distribute

A, f chosen to fit initial conditions: x(0) = x0 and v(0) = v0 PH421:Oscillations F09 4/17/2017 x m k A, f chosen to fit initial conditions: x(0) = x0 and v(0) = v0 Square and add: Divide: Please do not distribute

(A, f) because 2nd order linear differential equation PH421:Oscillations F09 4/17/2017 2 arbitrary constants (A, f) because 2nd order linear differential equation Please do not distribute

Position: Velocity: Acceleration: PH421:Oscillations F09 4/17/2017 Position: • A, f are unknown constants - must be determined from initial conditions • w0, in principle, is known and is a characteristic of the physical system Velocity: Acceleration: This type of pure sinusoidal motion with a single frequency is called SIMPLE HARMONIC MOTION Please do not distribute

PH421: Oscillations; do not distribute THE LC CIRCUIT DIFFERENTIAL EQUATION 11/12/07 L C I q Kirchoff’s law (not Newton this time) Same differential equation as the SHO spring! Lecture 5/6 - damped oscllations

What is inductance?? It is how much magnetic flux is created in the inductor coil by a given current I, in a wire. What is the voltage change across an inductor? A voltage change occurs WHEN there is change in magnetic flux (i.e. some of the energy is ‘converted’ to a magnetic field)

PH421: Oscillations; do not distribute THE LC CIRCUIT 11/12/07 L C I q Kirchoff’s law (not Newton this time) Lecture 5/6 - damped oscllations