Simple Pendulum Motion Derived Using Torque and the Small Angle Approximation
Summary
TLDRThis educational video script discusses the physics of a pendulum, drawing parallels with a mass on a spring system. It introduces torque and rotational motion concepts, using the rotational form of Newton's second law to derive the pendulum's equation of motion. The script simplifies the analysis with the small angle approximation, equating sine theta to theta, leading to a solvable differential equation. It concludes by highlighting the pendulum's period independence from mass and its dependence on length and gravity, providing insights into pendulum motion.
Takeaways
- 🔄 The lecture explains similarities between the pendulum system and the mass-on-spring system.
- ⚖️ Focus on analyzing forces acting on a swinging pendulum, including gravity and tension.
- 🌀 The lecturer uses the rotational form of Newton's second law (torque = I * alpha) to analyze pendulum motion.
- 📐 Torque is calculated using the effective perpendicular component of gravitational force (mg * sin(theta)) times the length of the string.
- ⚠️ A negative sign is included in the torque equation to indicate that the force opposes the increase in angle (theta), similar to a spring restoring force.
- ⏳ The lecture emphasizes that for pendulums, unlike with mass-spring systems, the motion is not dependent on the mass but on the length of the string.
- 💡 The second derivative of the angle (theta) relates to the sine of theta, making it a more complex differential equation than that of the mass-spring system.
- 📉 To simplify, the small angle approximation is introduced, treating sin(theta) as approximately theta for small angles.
- 🔁 Using the small angle approximation, the pendulum's motion equation resembles the mass-spring system, with the period dependent on the length of the string and gravity (T = 2*pi*sqrt(L/g)).
- 📊 The small angle approximation is valid as long as the angle remains small, supported by a Taylor series expansion and graph comparison.
Q & A
What is the primary difference between the motion of a mass on a spring and a pendulum?
-The motion of a mass on a spring is dependent on the mass, whereas the motion of a pendulum is not dependent on the point mass. In the equations of motion for the pendulum, the mass cancels out.
Why does the script mention the rotational version of F=ma?
-The script mentions the rotational version of F=ma, torque equals I*alpha, to analyze the pendulum's motion because pendulums involve rotational dynamics rather than linear motion.
What is the significance of the angle theta in the context of the pendulum's motion?
-Theta represents the angle between the force of gravity and the line of the string. It is crucial for determining the component of the gravitational force that contributes to the torque, which is essential for the pendulum's swinging motion.
Why is the torque equation written with a negative sign?
-The negative sign in the torque equation ensures that as the angle theta increases, the force tends to bring the pendulum back towards smaller angles, simulating the restoring force in a pendulum's oscillation.
What is the small angle approximation, and why is it used in the script?
-The small angle approximation is an assumption that sine(theta) is approximately equal to theta when theta is small. It simplifies the differential equation of the pendulum's motion, making it easier to solve and analyze.
How does the period of a pendulum relate to its length and the acceleration due to gravity?
-The period of a pendulum is given by T = 2*pi*sqrt(l/g), where l is the length of the pendulum and g is the acceleration due to gravity. This shows that the period depends on the length of the pendulum and the strength of gravity but not on the amplitude or the mass of the pendulum.
What is the role of the tension in the string during the pendulum's motion?
-The tension in the string acts to support the weight of the pendulum bob and maintain the circular path of the pendulum's motion. However, it does not contribute to the torque that causes the pendulum to swing back and forth.
Why does the script suggest that the pendulum's motion is not dependent on the amplitude?
-The script suggests that the pendulum's motion is not dependent on the amplitude because the equations of motion for the pendulum do not include the amplitude. The period of the pendulum is constant for small oscillations, regardless of the amplitude.
How does the script justify the use of the small angle approximation?
-The script justifies the use of the small angle approximation by showing that for small angles, the sine function can be approximated as linear, which simplifies the differential equation and allows for an easier comparison with the motion of a mass on a spring.
What is the significance of the moment of inertia (I) in the pendulum's motion?
-The moment of inertia (I) is significant in the pendulum's motion because it relates to the rotational inertia of the pendulum bob. It is used in the torque equation to determine the angular acceleration, and for a point mass, it is given by I = m*r^2, where m is the mass and r is the radius of rotation.
Outlines
🔍 Analyzing the Pendulum's Motion
The paragraph introduces the concept of analyzing the motion of a pendulum by comparing it to the motion of a mass on a spring. The speaker emphasizes the similarities between the two systems and starts by identifying the forces acting on the pendulum, such as gravity pulling it down and tension in the string. The speaker then transitions into using rotational dynamics, specifically torque, to analyze the pendulum's motion. The torque is calculated as the perpendicular component of the gravitational force times the length of the string. The goal is to establish a relationship between torque and angular acceleration, leading to an equation that parallels the force-mass-acceleration relationship seen in the mass-spring system. The speaker also discusses the importance of considering the negative sign to ensure that the restoring force acts to bring the pendulum back to equilibrium.
📐 Solving the Pendulum's Equation with Small Angle Approximation
This paragraph delves into the complexities of solving the differential equation for the pendulum's motion. The speaker notes that the sine function in the equation makes it challenging to solve, leading to the introduction of the small angle approximation. This approximation simplifies the sine function to the angle itself for small values of the angle, making the equation solvable. The speaker then uses the previously analyzed mass-spring system as a reference to find solutions for the pendulum's angular position, velocity, and acceleration. The period of the pendulum's motion is derived, highlighting its dependence on the length of the string and the acceleration due to gravity, but not on the mass of the pendulum. The speaker emphasizes that the small angle approximation is a simplification that works well for small angles but breaks down as the angle increases.
📉 Visualizing the Small Angle Approximation
The final paragraph provides a visual and mathematical explanation of the small angle approximation. The speaker explains that by using a Taylor series expansion or by graphing the sine function, it's evident that for small angles, the sine function can be approximated as linear, which corresponds to the angle itself. This approximation is valid as long as the angle remains small, as the higher-order terms become negligible. The speaker also points out that this approximation is what allows the pendulum's motion to be described using simple harmonic motion equations, similar to those of the mass-spring system. The paragraph concludes by illustrating the limitations of the small angle approximation as the angle increases and deviates significantly from linearity.
Mindmap
Keywords
💡Pendulum
💡Gravity
💡Torque
💡Moment of Inertia (I)
💡Angular Acceleration (alpha)
💡Small Angle Approximation
💡Differential Equation
💡Period
💡Amplitude
💡Taylor Series
Highlights
Introduction of the pendulum case to compare with the mass on a spring system.
Gravity acting on the pendulum is represented as mg, with an angle θ indicated.
Tension in the string is identified as an opposing force to gravity.
Use of the rotational version of Newton's second law, torque equals Iα, to analyze pendulum motion.
Identification of the torque as the product of the effective force and the string's length.
Explanation of the effective force as the perpendicular component of gravity, mg sine θ.
Inclusion of a negative sign to represent the restoring force that brings the pendulum back to equilibrium.
Moment of inertia I is defined as ml² for a point mass on the end of a string.
Derivation of the differential equation for the pendulum's angular motion.
Observation that pendulum motion is independent of the mass m, unlike the mass-spring system.
Simplification of the differential equation using the small angle approximation.
Derivation of the period of the pendulum using the small angle approximation.
Comparison of the pendulum's period with that of the mass-spring system, highlighting the independence from mass.
Graphical representation of the sine function to illustrate the small angle approximation.
Explanation of the Taylor series expansion of sine θ to justify the small angle approximation.
Final equation for the pendulum's angular motion as a function of time, θ(t).
Derivation of angular velocity and acceleration from the angular motion equation.
Transcripts
so next we're going to move it on and
look over at the at the case of the
pendulum
and i've purposefully left some notes
about uh movement of a mass on the
spring up here because i want you to be
able to see the similarities between the
the two systems so much like we did with
the mass on the spring to start to
analyze how this is going to move
um we want to put the forces down that
are acting on the object
that's going to swing back and forth so
i'm going to fixate on it while it's on
this
in this position what's going to happen
is you'll have gravity pulling
straight down like this so i'm going to
call the force of gravity mg
something that's going to help us later
is i'm going to also indicate this angle
between that force and the line of the
string
and then also pulling up this way is
going to be the tension
in the string so what we did with the
mass on the spring was we started with f
equals ma
it's possible to do that in a way with
this but what i'm going to choose to do
instead
is the rotational version of f equals ma
and so what i'm going to do instead
is i'm going to look at torque equals i
alpha so
instead of force equals mass times
acceleration i'm going to kind of do
like rotational force or torque
equals rotational mass or moment of
inertia
times well rotational acceleration or
angular acceleration
which is alpha so some of the torques is
i alpha
well so what we want to do is look at
how much torque there is on the system
uh here's the axis and so
torque we want to have uh the effective
force times the distance
oh it looks like i better give this
string a length let's call it l
so the string is going to have length l
for torque right what we want is
only the component of force that's
perpendicular to the string right so we
have kind of like a
a component of mg that would be this way
that i could call like mg parallel
that's not our guy that's not going to
make um
any torque we want the part that goes
this way
which would be like the perpendicular
part of mg
right so that would be the part the
component that's this way
which you can see that's kind of
opposite the angle we know so it would
be like mg
sine theta okay so the effective force
is going to be mg sine theta
so on the left hand side here we'll go
mg sine theta
for the effective force times the length
because you need force times distance
and then the other thing we need to
remember here is we want to put a
negative sign because
we want the as we increase theta we want
the force to tend to bring it back
towards smaller angles
if we neglected the negative sign that
would mean that the greater the angle we
pulled to the more it would want to go
away
so the thing would just fly away that's
kind of like with the mass on the spring
um you want to have the force be
negative kx because
as x gets bigger you want there to be
more incentive for it to come back
um just like there is with the with the
pendulum here so here's the torque
uh force times distance um equals i
alpha well i is going to be the moment
of inertia of a point mass
just mr squared or m times distance
squared
so that's m l squared
right and then times alpha now
i could put and so what alpha is um
i could write alpha which is the angular
acceleration but then what would happen
is i would have
theta which is changing in time and a
variable alpha which is changing in time
um so kind of too many variables too
many things changing at once
so what i would rather do is write it in
terms of just one variable
well so alpha right is the angular
acceleration so it's the second
derivative
of theta um so sorry the writing is
getting small here i'm going to soon get
out of this corner
uh d squared theta dt squared
right so again you can kind of see we
were at the same point here with
mass on the spring we had minus kx as m
d squared x dt squared
um here we have a pretty similar thing
getting getting built up
so what i'm going to do is exactly what
i did before which is to solve for the
the second derivative of the position
variable basically
the second derivative of the angle so
let's solve for that
looks like one of the lengths
goes away here looks like the mass goes
away
so that's key that's something that's
different from the
um mass on the spring here the motion
was dependent on the mass
uh with the pendulum the motion is is
not going to be dependent on the
on the point mass that you put here so
that's going to cancel out
and so just rearranging that a bit it
looks like we get
d squared as a matter of fact i'm going
to write it
i'm going to write it over here so it
looks just like what we had at this
stage
uh last time with the mass and the
spring d squared theta by dt squared
equals
you're going to get minus g over l sine
theta
okay so i'm gonna pause here for a
second because it's a big deal if you
can kind of look back and compare with
what we did before
notice that these so these are a couple
constants
these are a couple constants our
variable here was x so here
our variable's theta you can see these
are the same
equations except that
there is the added complexity of this is
the sign of the angle
okay this ends up being a much tougher
differential equation to solve
here it's just saying two derivatives of
the
position function give you the position
function back with a negative constant
front
here it says two derivatives of theta of
t
will give you the sine of that function
back
with a negative constant in front that's
much tougher to solve
um so partly be for that reason
um the physicist will will tend to make
what's called the
small angle approximation in other words
wouldn't it be nice if
the sine if it weren't sine theta if it
were just theta
because then we'd have the exact
equation that we have here
and so what we are going to do is make
that approximation
and we'll say that if theta is small
i'll put that in quotes okay
then what will happen is sine theta is
pretty much
just theta right now you can check that
out if you just pull out a calculator
and play around in radians
i may make another side little video
about that um
my students are often pretty triggered
by this because this is like i thought
physics dealt with the truth and you're
lying now
what we're going to do is just making an
approximation called the small angle
approximation
and you get d squared theta by dt
squared
equals minus g over l theta
if you just roll with me on that we can
come come back to uh this thing later
um let's just see what happens if we if
we go for it with this well
if this is the case if you can actually
stomach this for the time being
we can cheat off of our previous work
and then just write down the solutions
this says there's a function out there
if i take its derivative twice
i get it back with a negative constant
in front well we already did that with
the mass on a spring
right so we can kind of cheat off the
work we did before and say well instead
of being x of t it's theta of t
is going to be some kind of an amplitude
times cosine
of well instead of root k over m now
it's going to be root g over l
times t plus a phase
okay so that will be the solution to the
um for the angle as a function of time
you can get the
angular velocity by taking one
derivative and the angular acceleration
by taking two derivatives
you'll see if you take two derivatives
of this function it will actually work
in here
the next thing we can do to cheat off of
our previous work is you notice
we had period as 2 pi root m over k
well we have the exact same equation but
with different critters different
constants
and so we can just cheat off of our
previous answer
and see that we get 2 pi root now
instead of it being
m over k it's going to be l over g in
this case
right so now you notice the period
is 2 pi root l over g to tie back with
what we did before
another skull and crossbones here
it obviously depends on the length and
on how strong gravity is
but again there is no dependence on the
amplitude
amplitude
and then to look back at what we did
before here
or the mass notice the mass doesn't show
up at all
in the um in the equations of motion for
the
for the pendulum if you increase the
mass you would have more torque from
um because of the increased weight
but then you'd also have a greater
moment of inertia and so those effects
kind of compete and it comes out in the
wash so
it ends up not influencing them the
motion okay
as far as the small angle approximation
goes
um one way to see um why we could maybe
get away with this
is if you've had enough math to know
about um
to know about uh taylor series um
sine theta you can write as as an
infinite series that goes like
theta minus
theta cubed over three factorial plus
theta to the fifth over five factorial
and so if you neglect if theta is really
small let's say
then theta cubed would be super tiny and
theta to the fifth will be really tiny
um
you're what you're basically doing is
neglecting these like higher order terms
and so you're just kind of cherry
picking out this
um this first term another way to see it
is simply make a graph of the sine
function
so here's a sine theta let's say against
theta
um the sine function of course looks
looks like this
all we're saying is that if you cherry
pick and kind of go for like small
angles here
if you don't let the angle get too big
what it's saying is that sine
theta is pretty much just theta or it
approaches a line
um and so for for the as long as you
kind of stay within this regime
um in this early part of the graph
um what we're doing is we're saying sine
theta equals theta as
as long as the angle's small obviously
that it gets terrible as the angle gets
big you can see the line deviating from
the sine graph quite a bit
so that's what's called the small angle
approximation and that is how a pendulum
moves
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