Flipping Physics Lecture Notes: AP Physics C: Simple Harmonic Motion Review (Mechanics) https://www.flippingphysics.com/apc-simple-harmonic-motion-review.html • An object is in Simple Harmonic Motion if the acceleration of the object is proportional to the object’s displacement from an equilibrium position and that acceleration is directed toward the equilibrium position. a ∝ Δx • For example: A horizontal mass-spring system on a frictionless surface has the following free body diagram: k o F F ma kx ma a x ∑ x = − s = x ⇒ − = x ⇒ x = − m o Amplitude, A, is defined as the maximum distance from equilibrium position. Therefore: k § amax = A m dv dx d ⎛ dx ⎞ d 2 x o Note: a = & v = ⇒ a = = dt dt dt ⎜ dt ⎟ dt 2 ⎝ ⎠ d 2 x k Therefore: o 2 = − x dt m k 2 o Let = ω where ω is called the angular frequency m 2 d x 2 o Therefore: = −ω x dt2 § This is the condition for simple harmonic motion. § This equation is not on the AP equation sheet. Memorize It!! k Δθ 2π m o Note: ω = = = ⇒ T = 2π (The period of a mass-spring system) m Δt T k L o Period of a pendulum: T = 2π (know how to derive) g 1 2π o T = & ω = = 2π f ⇒ ω = 2π f f T § Frequency, f, is the number of cycles an object goes through per second. § Angular frequency and frequency are related, ω = 2π f , however, they are not the same. 0204 Lecture Notes - AP Physics C- Simple Harmonic Motion Review (Mechanics).docx page 1 of 3 • One equation that satisfies the condition for Simple Harmonic Motion is: x (t) = Acos(ωt +φ) o This equation is on the AP physics equation sheet, however, the equations for velocity and acceleration in simple harmonic motion are not. o Have to use angles in radians in this equation. o φ or “phi” is the “phase constant” or the “phase shift” of the wave. For example: ⎛ π ⎞ § y = cos θ + is “phase shifted” to the ⎝⎜ 2⎠⎟ π left from y = cosθ by radians. 2 dx d • v = = Acos(ωt +φ) = A −sin(ωt +φ) (ω ) dt dt ( ) ( ) d ⎡ d ⎤ o ⇒ v t = A cos ωt +φ = A −sin ωt +φ ωt +φ ( ) dt ( ( )) ( ( ))⎢dt ( )⎥ ⎣ ⎦ o ⇒ v t = − Aω sin ωt +φ ( ) ( ) & o vmax = Aω dv d d ⎡ d ⎤ • a = = − Aω sin ωt +φ = − Aω sin ωt +φ = − Aω cos ωt +φ ωt +φ dt dt ( ( )) dt ( ( )) ( )⎢dt ( )⎥ ⎣ ⎦ 2 o ⇒ a = − Aω (cos(ωt +φ))(ω ) ⇒ a(t) = − Aω cos(ωt +φ) 2 & o amax = Aω 2 2 2 d x 2 • ⇒ a t = −ω Acos ωt +φ = −ω x t ⇒ = −ω x ( ) ( ( )) ( ) dt2 0204 Lecture Notes - AP Physics C- Simple Harmonic Motion Review (Mechanics).docx page 2 of 3 • Simple Harmonic Motion is NOT Uniformly Accelerated Motion • Total mechanical energy in Simple Harmonic Motion: 2 1 2 1 o ME kA m v total = = ( max ) 2 2 0204 Lecture Notes - AP Physics C- Simple Harmonic Motion Review (Mechanics).docx page 3 of 3 .