Simple Harmonic Motion AP Physics C

Simple Harmonic Motion What is it?  Any periodic motion that can be modeled with a sin or cosine wave function.  Harmonic oscillators include: Simple pendulum – a mass swinging on a string or rod Mass on a spring that has been offset from its rest position and then released

Reminders: Hooke’s Law and Conservation of Energy Hooke’s Law Elastic Potential Energy Conservation of Energy

Waves are oscillations too! Remember Wave Characteristics Amplitude  the maximum displacement of the medium measured from the rest position. Wavelength  The distance between corresponding points on consecutive waves. Frequency  the number of complete cycles (waves) that pass a given point in the medium in 1 second. Period  the time it takes for one complete cycle to pass a given point in the medium, or the time that passes before the motion repeats itself Wave Speed  velocity the wave travels through a medium

Simple Harmonic Motion (SHM) velocity & acceleration a=max Position (for reference only) A B C D E v=0 v=max v=0 v=max a = 0 At t=0: X = A (max disp), F=kA (toward x=0), a=kA/m (max), v=0, U el =1/2 kA 2, K=0. a=max Note: k= spring constant K=kinetic energy, T=period At t=T/4: X = 0 (no disp), F=0 (at equilibrium), a=0 (no force), v=max, U el =0 (no stretch), K=1/2 mV 2. At t=T/2: X = A (on other side of x=0), F=kA (toward x=0), a=kA/m (max), v=0, U el =1/2 kA 2, K=0. At t=T: Back to original position so same as t=0, X = A (max disp), F=kA (toward x=0), a=kA/m (max), v=0, U el =1/2 kA 2, K=0. At t=3T/4: X = 0 (no disp), F=0 (at eq.), a=0 (no force), v=max (opp. dir.), U el =0 (no stretch), K=1/2 mv 2.

More velocity & acceleration in SHM What happens between x=0 and x=A? (t=0 to t=T/2) a=max v=0 x=A Between t=0 and t=T/4, the mass is moving toward the equilibrium position (from x=A to x=0) with a decreasing force. a, v and a in same dir so v. U el, and K. Between t=T/4 and t=T/2, the mass is moving away from the equilibrium position (from x=0 to x=A) with an increasing force. a, v and a in opp dir, so v. U el, and K a v=max v v a Note: F & a are always directed toward x=0 (eq)

More velocity & acceleration in SHM What happens between x=0 and x=A? (t=T/2 to t=T) x=A a=max a=0 a a v v Note: F & a are always directed toward x=0 (eq) Between t=T/2 and t=3T/4, the mass is moving toward the equilibrium position (from x=A to x=0) with a decreasing force. a, v and a in same dir so v. U el, and K. Between t=3T/4 and t=T, the mass is moving away from the equilibrium position (from x=0 to x=A) with an increasing force. a, v and a in opp dir, so v. U el, and K.

Angular Frequency For SHM we define a quantity called angular frequency, ω (which is actually angular velocity), measured in radians per second. We use this because when modeling the SHM using a cosine function we need to be able to express frequency in terms of radians.

Modeling SHM with a cosine wave. When we start an oscillation, such as a mass on a spring, we either stretch or compress a the spring a certain distance which then becomes the Amplitude of the oscillation. Where A=Amplitude, ω=angular frequency, t=time, and φ = phase shift Note: when t=0, x=A (max displacement)

Modeling velocity for SHM Notice that at t=0, v=0 and this corresponds to the maximum displacement (x=A). Also…the maximum value occurs at sin(∏/2)=1, so v max =-ωA

Modeling acceleration and finding maximum acceleration Notice that at t=0, a=max in the opposite direction as the stretch and this corresponds to the maximum displacement (x=A). Also…the maximum value occurs at cos(0)=1, so a max =-ω 2 A

Calculating acceleration for a mass on a spring This means that acceleration is a function of position. Or… more stretch means more acceleration. We will express acceleration in a general for in terms of angular frequency, ω, and position, x, as shown. Note: at x=A, a=max = -ω 2 A

Simple Pendulum A simple pendulum is an object of mass, m, swinging in a plane of motion, suspended by a massless string or rod. In other words, it is a point mass moving in a circular path. l l θ If θ < 10 °, then we can assume a small angle approximation, sin θ ≈ θ, the long formula for period, which includes an infinite sine series, reduces to… Note: