Flipping Lecture Notes:

AP Physics C: Simple Harmonic Review () https://www.flippingphysics.com/apc-simple-harmonic-motion-review.html

• An object is in Simple Harmonic Motion if the of the object is proportional to the object’s from an equilibrium and that acceleration is directed toward the equilibrium position. a ∝ Δx • For example: A horizontal - 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 , 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 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 : 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. § and frequency are related, ω = 2π f , however, they are not the same.

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• 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 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 . 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

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• Simple Harmonic Motion is NOT Uniformly Accelerated Motion

• Total mechanical in Simple Harmonic Motion: 2 1 2 1 o ME kA m v total = = ( max ) 2 2

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