Simple Harmonic Motion Flashcards
(33 cards)
Periodic Motion
a motion which repeats after a finite amount of time
equation that proves when a motion is periodic??
x (t + T) = x(t)
same with v instead of x (velocity vs position)
if T is time period and t is actual time spent, means will return to same position and velocity regularly/PERIODICALLY!! <3
oscillation
complete back and forth
frequency!! freak quency
oscillations per time period, given by
f = 1/T
measured in s ^-1 or Hz, 1 Hz = one oscillation /second
T
period/time to complete one oscillation
general notes on sin function
begins at origin, 0
period of 2pi unless changed
amplitude of 1 unless changed
up first
o with line through it would be 3pi/4
general nots on cosine
begins at y = 1
period of 2pi unless changed
amplitude of 1 unless changed
down first
o with line through it would be 0
manipulation of cosine and sin
[A] cos/sin(x) - og amplitude would be one, this is multiplied by A, if A> 1 stretch, <1 compression. If A is negative, graph is flipped along the x axis!
cos/sin ([n] x) - horizontal stretch/compression, og wavelength is 2pi, if any number before x etc, divide 2pi by that to get new wavelength!! ex. sin (2x), 2pi/2 = pi therefore new wavelength is pi. IF n is -, technically flip over the y axis but same results as flip over x
cos/sin (x +/- g) - shift left or right by g units, COUNTERINTUITIV`E if (x +…) shift to the left! if (x - ….) shift to the right
cos/sin (x) + h - shift whole ass graph up or down by h! actually makes sense, + h shift up so midpoint is 0 + h, if - h shift down so midpoint is 0 - h
relationship between sin and cos, how to transform into the other
sin (theta + pi/2) = cos(theta)
cos (theta - pi/2) = sin (theta)
what the heck does multivalued mean???
multiple values of theta/ angle inserted into sin/cos equation will give the same results
IMPORTANT equation for simple harmonic motion!!!
x(t) = A cos (wt + [o with line down center])
A = amplitude
w = angular frequency (calculated by 2pif or 2pi/T) f is frequency i think confirm this
[o with line down middle] = phase constant (determined by initial position and velocity of graph, related to position on the unit circle
how to calculate fancy w???
2pi/T OR 2pif where T is the period and f is frequency because remember frequency = 1/T
units = radians
how to determine phase constant
fancy o with line (phase constant) = cos^-1 ([x(0)] / A)
x(0) is y position at x = 0
how to find velocity of shm
take the derivative of position function!!!
v(t) = dx(t)/d(t) = d/dt (Acos (wt + funky circle)
v(t) = -(wA) sin (wt + funky circle)
calculate max velocity???
vmax = wA OR (2pi/T x A)
derivative of cosx
-sinx! if Acos(wt + h)
derivative is -(Aw) sin (wt + h)
derivative of sinx
cosx!!!
what is velocity when max displacement???
0!!
what is displacement when velocity max?
ZERO
how do you find acceleration?
derivative of velocity equation!
d/dt (-wA sin (wt + funky circle))
= -(w^2 * A) cos (wt + funky circle!)
how do you calculate maximum acceleration??
w^2 * A
relationship between acceleration and displacement
always exactly opposite, literally just flip the graph (with different ampitudes)
relationship between acceleration and velocity
when the slope of velocity is positive (ex moving from -A to +A), graph of acceleration is positive
when slop velocity negative (+A to -A), graph of acceleration is negative
relationship between force and SHM
if direction of force is always opposite to direction of the displacment the motion will be simple harmonic motion
