Mr. P.: Good morning.
Previously we derived a position equation for an object moving in simple harmonic motion.
♫ Flipping Physics ♫
Mr. P.: Now we are going to derive equations for velocity and acceleration
of objects moving in simple harmonic motion.
Bobby, remind me, what is the equation we previously derived
for the position of an object in simple harmonic motion?
Bobby: Position as a function of time equals amplitude times cosine of
2 pi times frequency times time.
Mr. P.: Yes, this is the equation given on the AP Physics 1 equation sheet.
But, in order to derive velocity and acceleration as functions of time,
I would like this equation in terms of angular frequency instead of frequency.
Bo, could you please derive the equation for angular frequency?
Bo: Isn't that the one which is pretty much already on the AP Physics equation sheet?
Mr. P.: Yes, but I still need you to derive it, please.
Bo: Sure.
Bobby: Remember, you start with the equation for angular velocity.
Bo: Oh yeah... angular velocity equals change in angular position over change in time,
for one revolution change in theta equals 2 pi radians and change in time is the period,
period and frequency are inverses of one another,
therefore omega, or angular frequency, equals 2 pi times frequency.
Mr. P.: Therefore the position of an object in simple harmonic motion can be described by the equation
position equals amplitude times the cosine of angular frequency times time.
Plus, I have added the phase constant phi, or the phase shift,
into the equation to make the equation more generally applicable.
Bobby: Phi? Billy: The phase constant?
Bo: Remember, phi, the phase constant,
it shifts the sine or cosine curve along the x axis, that's all it does.
Billy: Yeah, that's right, sine and cosine both have the same shape,
it is just that sine is phase shifted to the right from cosine by 90 degrees or pi over 2 radians, thanks.
Bobby: Right.
Mr. P.: Okay, now that we have the position equation, let's work on deriving the velocity equation.
Bobby, please remind me, how are position and velocity related?
Bobby: Well, velocity equals change in position over change in time, right?
Mr. P.: Right, but I'm looking for the calculus version.
Bobby: Oh.
Billy: Well, velocity equals the derivative of position with respect to time.
Mr. P.: Correct, now, I know some of you might not have taken calculus yet
and might not understand derivatives.
Realize derivatives are needed in order to derive
velocity and acceleration of simple harmonic motion equations.
If that's true, then this math will go over your heads. However, I still think it's useful because
it's really good to get some exposure to.
Billy, please take the derivative with respect to time of our position equation.
Billy: Well, in order to take the derivative with respect to time of
amplitude times the cosine of angular frequency
times time plus the phase constant, we need to use the chain rule.
Amplitude is constant, so just leave that up front,
the derivative of cosine is negative sine, so we have negative sine of angular frequency
times time plus the phase constant, and we need to multiply by the derivative
with respect to time of the angular frequency times time plus the phase constant,
which works out to be just angular frequency.
So our simple harmonic motion equation for velocity with respect to time equals
the negative of the amplitude times angular frequency times the sine of angular frequency times time
plus the phase constant.
Mr. P.: Great. So, that is an equation for the velocity of an object in simple harmonic motion,
and because the value of sine has a maximum of 1 and a minimum of -1,
the maximum value for the velocity of an object in simple harmonic motion equals amplitude times
angular frequency.
Bo, please derive our corresponding equation for the acceleration of an object in simple harmonic motion.
Bo: Acceleration equals the derivative of velocity with respect to time,
so we need to take the derivative of the velocity equation Billy just derived.
Amplitude and angular frequency are constants so they can go out in front of the derivative,
the derivative of sine is cosine,
and again, we multiply by the derivative of omega t plus phi, which is omega.
So, our acceleration as a function of time for an object in simple harmonic motion
equals negative amplitude times angular frequency squared
times the cosine of angular frequency times time plus the phase constant.
Mr. P.: Yes, and again, because the value of cosine theta varies between -1 and 1,
the maximum value for acceleration equals amplitude times angular frequency squared.
And going back to the position equation, we can use the same rationale to show that the maximum position
equals the amplitude. So notice, the maximum position equals the amplitude,
the maximum velocity equals the amplitude times the angular frequency,
and the maximum acceleration equals the amplitude times the angular frequency squared.
One more quick thing,
I just want to remind everyone that because the derivations
of these equations requires theta to be in radians,
all angles in these equations need to be in radians,
and your calculator needs to be in radian mode
when using equations for position, velocity, and acceleration
as a function of time in simple harmonic motion.
Billy: Oh, thank you for the reminder, Mr. P.
Mr. P.: You are welcome.
Thank you very much for learning with me today, I enjoyed learning with you.