Good morning. We have already derived simple harmonic motion equations using a mass-spring
system; today we are going to derive the equations for simple harmonic motion of a simple pendulum
using calculus. Calculus!
♪ Flipping Physics ♪ Again?
Bo, what is the equation for the period of a simple pendulum?
I thought we were deriving that. (We are, however, we have already worked with the
equation, so I thought we would review it first.) Sure. The period of a pendulum equals two pi times
the square root of the quantity length of the pendulum over little g, the gravitational field
strength. And this equation is only valid if the pendulum angle is less than 15 degrees.
I thought little g was the acceleration due to gravity.
Gravitational field strength and acceleration due to gravity are the same thing.
Oh, right. Thank you Bo. Bobby, what is the general
equation which defines simple harmonic motion? Oh, we went over this last time.
The second derivative of position with respect to time equals the negative of the square of
angular frequency times position where angular frequency equals the square root of the quantity
spring constant over mass. Angular frequency only equals
the square root of spring constant over mass for a mass-spring system.
Oh right. The angular frequency is different for each type of system we are analyzing. So,
it’ll be different for a pendulum. Yeah. That is correct. What we are doing today
is deriving the angular frequency of a simple pendulum so we can use that to derive the period
of a simple pendulum. Billy, please remind me what the restoring force for a pendulum is.
The restoring force for a pendulum. … Uh, but,
have we defined what a simple pendulum is yet? Ah, good point Bo. We’ve talked about it a bit;
however, you are correct that we should specify exactly what it means to be a simple pendulum.
So, Billy, what is a simple pendulum? A simple pendulum? Aren’t all pendulums simple?
No, all the pendulums we have worked with so far have been simple pendulums, however,
most real life pendulums are not actually simple pendulums. A simple pendulum is
basically an idealized, simple version of a pendulum. Hence “simple pendulum”.
Okay. So, for example, we have assumed the string of each of our pendulums has negligible mass.
Is that part of what it means to be a simple pendulum?
(Yes, exactly. What else?) Uh …
We have assumed the pendulum bob is a point mass,
meaning it has no shape and zero size. And we have assumed there is no friction,
either with the medium the pendulum bob is moving through or between the string
and whatever it is attached to at the very top. Exactly. A simple pendulum assumes the string
has mass small enough relative to the mass of the pendulum bob that the mass of the
string is negligible, the pendulum bob is a point mass, and there is no friction.
One other assumption about simple pendulums is that the string is inextensible.
Inextensible? Incapable of being extended.
Yeah, the length of the string will never change. Alright, we are back to asking Billy what
the restoring force for a pendulum is. Billy? The restoring force for a pendulum. Well,
a restoring force is a force which is always directed towards equilibrium and proportional
to distance from equilibrium. For a pendulum that is the component of the force of gravity on the
pendulum bob which acts tangent to the direction of the motion of the pendulum.
Do you want me to solve for the equation for force of gravity tangential?
(Yes Billy. That would be great. Thanks.) Well, the angle for a pendulum is between
the string and the vertical. That is the same as the angle
in the vector triangle where we break force of gravity into its components in the tangential
and out directions. So … (SOH is correct Billy.)
Nice. Sine of theta equals opposite over hypotenuse. Opposite theta is the force of
gravity tangential and on the hypotenuse is the force of gravity. Therefore,
the force of gravity tangential equals the force of gravity times sine of theta or mass times
gravitational field strength times sine of theta. Thanks Billy. Now we can sum the forces in the
tangential direction. That equals the positive of the force of gravity tangential which also equals
mass times acceleration tangential. Substituting in the negative of the equation Billy solved for
gives us the negative of mass times gravitational field strength times sine
theta equals mass times tangential acceleration. Hold up. Where did that negative come from?
We did this last time. Because the force of gravity tangential is always directed opposite the
displacement from equilibrium it is a restoring force. That means the tangential acceleration and
angular displacement theta are always in opposite directions. That’s where the negative comes from.
No matter which side of equilibrium the pendulum bob is on,
the angular displacement and acceleration are always in opposite directions. Thanks.
Yes, thanks. Substituting in the equation Billy solved for gives us the negative of mass times
gravitational field strength times sine theta equals mass times tangential acceleration.
Now, realize tangential acceleration is the second derivative of position with respect to time,
however, this is not linear motion, this is motion along an arc,
therefore, this position is arc length. Tangential acceleration equals the second
derivative of arc length with respect to time. And Everybody brought mass to the party!
Everybody brough mass to the party! Now, arc length equals radius times
angular position. The radius for the motion of the pendulum is the length of the pendulum, capital L.
And remember the length of the pendulum is the distance from the center of suspension
to the center of mass of the pendulum, not just the length of the string.
That is correct Billy. We can use this equation to identify that an
infinitesimally small distance along the arc length ds equals the length of the pendulum
times an infinitesimally small displacement d theta. How is that helpful for our derivation?
That means the second derivative of arc length with respect to time
equals L times the second derivative of angular position with respect to time.
And we can divide by the length of the pendulum. We are so close to the simple harmonic motion
equation only we have theta instead of x. Yeah. We just need theta instead of sine
theta on the right-hand side of the equation. I feel like this is where we use the fact that
the simple pendulum period equation is only valid for pendulum angles that are less than 15 degrees.
Yeah… That is correct. This is where
we use the small angle approximation. The small angle approximation states that the sine of theta
and theta are approximately equal to one another for small angles.
Small angles? Who gets to decide if it's a small angle?
Right. That is a reasonable question. First off, realize the small angle approximation is
only valid when the angle is in radians. And let’s see how this holds up for an angle of 15 degrees,
the angle Bo mentioned earlier as the maximum angle for a simple pendulum to be considered
to be in simple harmonic motion. Bobby, please convert 15 degrees
to radians and take the sine of that angle. To convert an angle of 15 degrees to radians
we multiply by pi radians over 180 degrees. That gives us 0.2618 radians. And the sine of that
equals 0.2588 which is pretty close to 0.2618. Correct Bobby. At 15 degrees, when converted
to radians, there is roughly a 1 percent difference between the sine of the angle and
the angle. Typically, between 10 and 15 degrees as a maximum value for theta is where the small angle
approximation is considered to be valid. Just realize that the larger the angle, the larger
the error in this approximation, and definitely do not go over 15 degrees. Now we can substitute
theta in for sine of theta, and we see that we have an equation which satisfies the condition for
simple harmonic motion. The second derivative of angular position with respect to time
equals the negative of the gravitational field strength over period all times angular position.
Billy, please determine the angular frequency and period of a pendulum in simple harmonic motion.
Absolutely. Well, we know the condition for simple harmonic motion is the second derivative of
position with respect to time equals the negative of the square of angular frequency times position.
We can replace linear position with angular position. And the square of angular frequency
equals gravitational field strength, little g, divided by pendulum length. So, angular
frequency equals the square root of gravitational field strength divided by pendulum length. We
also know period equals two pi divided by angular frequency; we derived that last time. Substitute
in the square root of gravitational field strength divided by pendulum length for angular frequency.
And we get that the period of a simple pendulum when the angle is less than 15 degrees equals two
pi times the square root of the quantity pendulum length divided by gravitational field strength,
the equation Bo gave us earlier. Very nice Billy. Thanks.
We can extend this to the equations for position, velocity, and acceleration as a function of time
for the pendulum as well. We already derived the linear equations using a mass-spring system.
Notice the derivation of the angular equations of motion for a simple pendulum are very similar,
we simply replace all of the linear variables with angular variables. For example, rather than linear
position as a function of time equals amplitude times cosine of the quantity angular frequency
times time plus phi, the phase constant. We have angular position as a function of time equals
maximum angular position times cosine of the quantity angular frequency times time plus phi.
We can do the same thing with the velocity and acceleration equations.
Please make sure you remember to use radians in all of these equations.
Realize that means amplitude, or theta maximum, for a simple pendulum is measured in radians.
And the equations for the maximum value of each of these works out to be the same as well,
position maximum, is just, well, position maximum. Velocity maximum equals maximum position times
angular frequency. Acceleration maximum equals maximum position times angular frequency squared.
And, for a simple pendulum, we know angular frequency equals the square root
of the quantity gravitational field strength over pendulum length. Thank you very much for learning
with me today, I enjoy learning with you. Uh, I have a question. mr.p?
Is he gone already? Can one of you help? Sorry, I have to get to math club.
(I guess I can help. What’s the question?) Well, does omega mean angular velocity or
angular frequency? What do you mean?
One of equations for the simple pendulum has omega in it three times and I think it means
two different things in the same equation. Angular velocity as a function of time
equals the negative of the maximum angular position times angular frequency times the
sine of the quantity angular frequency times time plus phi, the phase constant. Does omega really
mean two different things in the same equation? Yeah, I guess it does. Omega stands for the
changing angular velocity of the pendulum bob as it move back and forth as a function of time and
omega stands for the angular frequency of simple harmonic motion which is a constant number and
for a simple pendulum angular frequency equals the square root of gravitational field strength
over pendulum length. I guess omega does stand for two different things in the same equation.
Bummer. Yeah.