Good morning. We have already derived simple harmonic motion equations for a
mass-spring system, a simple pendulum, and a physical pendulum. Today, we are going to
derive the maximum mechanical energy in an ideal, horizontal mass-spring system.
In a mass-spring system the spring will have elastic potential energy,
and because, in an ideal mass-spring system, the spring has zero mass, the kinetic energy of
a mass-spring system is concentrated in the mass only. So, this horizontal mass-spring
system can have both kinetic energy in the mass and elastic potential energy in the spring.
Let’s start with just the elastic potential energy stored in the spring. Bobby, what is the equation
for elastic potential energy? ♪ Flipping Physics ♪
Elastic potential energy equals one half the spring constant times the square of x,
the displacement from equilibrium position. Thank you Bobby. Billy, what is an equation
we derived for the x-position of a mass-spring system as a function of time?
Position as a function of time equals amplitude times the cosine of the quantity angular frequency
times time plus phi, the phase constant. Thank you Billy. When we substitute that
position equation into the elastic potential energy equation, we get one half times the
spring constant times amplitude squared times the square of the cosine of the quantity angular
frequency times time plus the phase constant. Bo, what is an equation for the velocity of this mass
which is moving in simple harmonic motion and the equation for its kinetic energy?
We can take the derivative with respect to time of the position equation Billy gave us to get
velocity as a function of time equals the negative of amplitude times angular
frequency times the sine of the quantity angular frequency times time plus the phase constant.
Kinetic energy equals one half mass times velocity squared. We can substitute the
equation for velocity into the kinetic energy equation and we get that kinetic energy equals
one half mass times amplitude squared times angular frequency squared times the
square of the sine of the quantity angular frequency times time plus phase constant.
Very nice Bo, thank you. Now, the total mechanical energy of the mass-spring system
at any point equals the sum of the kinetic and elastic potential energies. We have equations
for both of those which we can substitute in and we now have an equation for the total mechanical
energy of the system at any point in time. That is just ugly and ungainly.
Yeah. It does look confusing.
Huh, well, let’s simplify it then. Bo, what is the angular frequency equation
for a mass-spring system? Uh, I don’t know.
(Just look in your notes from two lectures ago.) I don’t take notes.
I just look at the lecture notes you post at flippingphysics.com if I forget anything.
Nice. ( Yeah. Billy?)
The angular frequency of a mass-spring system equals the square root of the
quantity spring constant divided by mass. (Thank you Billy.) Notice then that we can
square that whole equation, and multiply through by mass, to get spring constant equals mass times
angular frequency squared. And mass times angular frequency squared is in the total mechanical
energy equation for the mass-spring system, which means we can substitute spring constant in for
mass times angular frequency squared. Anybody see anything we can do with that equation?
We can factor out one half spring constant times amplitude squared. But
I don’t see how that helps … We can use a trigonometric identity.
Sine squared theta plus cosine squared theta equals one. Everything between the brackets equals
one. That means the total mechanical energy of the mass-spring system at any point equals one half
times spring constant times amplitude squared. That actually makes sense. When the mass is
at the amplitude, or the maximum distance from equilibrium, the mass is at rest and has
zero kinetic energy, so all of the mechanical energy is stored in elastic potential energy
in the spring. Cool! Sure.
Very nice y’all. Now, the total mechanical energy of a horiztonal mass-spring system can also be
expressed in terms of kinetic energy when the mass is located at the equilibrium position.
Because the mass-spring system has zero elastic potential energy at this location,
all the mechanical energy when the mass is at the equilibrium position is kinetic energy.
In other words, the total mechanical energy of the system also equals one half mass time
maximum velocity squared. We can set those two total mechanical energies equal to one another.
Multiply through by 2. Divide by mass and take the square root of the whole equation to get
The maximum velocity of the mass spring system equals amplitude times the square
root of the quantity spring constant divided by mass. Class, does that look familiar?
Oh. Angular frequency equals the square root of the quantity spring constant divided by mass.
That means the maximum velocity of the mass spring system equals amplitude
times angular frequency. (That is true.
And have we seen maximum velocity equals amplitude times angular frequency somewhere before?)
Right. When we derived the motion equations for simple harmonic motion,
we showed that the maximum velocity in simple harmonic motion equals amplitude times angular
frequency. So, we just derived that equation again, in a completely different way. Cool.
Yeah, Cool! You can remember that
the maximum velocity in simple harmonic motion equals amplitude times angular frequency, but
you didn’t remember that the angular frequency of a mass-spring system equals the square root of the
quantity spring constant over mass? How is that? Actually, I did not remember this equation either.
Uh What? The maximum velocity of simple
harmonic motion is right there in the velocity as a function of time equation I derived earlier.
The value of the sine of any angle varies between negative one and one. So, the maximum value for
velocity equals amplitude times angular frequency. Okay. Right.
Oh. Yeah.
Thank you very much for learning with me today, I enjoy learning with you.