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:

Physical Pendulum pivot cm θ D D mg Dsinθ θ = angular displacement D = distance between center of mass and pivot. A physical pendulum is any object that is oscillating and does not have all of its mass concentrated at one point. Also, it does not oscillate about its center of mass, therefore it has a rotational inertia (I). Figure AFigure B In Fig. A the “blob” is hanging at “rest position”. In Fig. B the “blob” has been displaced to an amplitude of θ and will be released to oscillate from that position. When the mass is displaced from equilibrium the weight, acting at the center of mass causes a torque on the object, which in turn causes it to oscillate. We need to analyze the Torque. Let’s concentrate on the middle part. Assuming small angle approximation. If we compare the result for with a max found earlier, we see that they have similar forms. therefore…