Simple Pendulum Motion Derived Using Torque and the Small Angle Approximation

Summary

This 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

00:00

🔍 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.

05:00

📐 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.

10:02

📉 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.

Keywords

💡Pendulum

A pendulum is a weight suspended from a fixed point so that it can swing freely back and forth under the influence of gravity. In the video, the pendulum is used to illustrate the principles of rotational motion and compare them with those of a mass on a spring. The script discusses how the forces acting on a pendulum can be analyzed to understand its motion, focusing on the gravitational force and the tension in the string.

💡Gravity

Gravity is the force that attracts two objects towards each other, and it is the force that causes the pendulum to swing. In the script, gravity is described as pulling straight down on the pendulum, with the force denoted as 'mg', where 'm' is the mass of the pendulum bob and 'g' is the acceleration due to gravity. This force is essential for the pendulum's motion and is a key factor in the equations derived to describe its movement.

💡Torque

Torque is a measure of the force that can cause an object to rotate about an axis. In the context of the video, torque is calculated as the product of the force component perpendicular to the string and the length of the string ('mg sine theta times l'). This torque is crucial for understanding the rotational motion of the pendulum and is used in the rotational version of Newton's second law, 'torque equals I alpha', where 'I' is the moment of inertia and 'alpha' is the angular acceleration.

💡Moment of Inertia (I)

The moment of inertia is a measure of an object's resistance to rotational motion about a particular axis and is used in the equation relating torque to angular acceleration. For a point mass pendulum, the moment of inertia 'I' is given by 'm times l squared', where 'm' is the mass and 'l' is the length of the pendulum. This concept is central to the script's discussion of the pendulum's motion, as it relates to the conservation of angular momentum.

💡Angular Acceleration (alpha)

Angular acceleration is the rate of change of angular velocity with time, analogous to linear acceleration in straight-line motion. In the script, it is symbolized by 'alpha' and is derived from the rotational form of Newton's second law. The equation 'torque equals I alpha' is used to find the angular acceleration of the pendulum, which is essential for predicting its motion over time.

💡Small Angle Approximation

The small angle approximation is a simplification used when the angle of rotation is small enough that the sine of the angle can be approximated as equal to the angle itself. This approximation is used in the script to simplify the differential equation describing the pendulum's motion, leading to a more manageable form. The script explains that for small angles, 'sine theta' can be approximated as 'theta', which allows for easier calculation and understanding of the pendulum's behavior.

💡Differential Equation

A differential equation is an equation that relates a function to its derivatives, and it is used to describe how a quantity changes over time. In the video, the script derives a differential equation for the pendulum's motion, 'd squared theta by dt squared equals minus g over l sine theta'. This equation is central to understanding the pendulum's dynamic behavior and is used to find the period of the pendulum's swing.

💡Period

The period of a pendulum is the time it takes for the pendulum to complete one full swing (back and forth). In the script, the period is derived from the differential equation and is given by '2 pi root l over g'. This result shows that the period of a pendulum is independent of its mass, a key insight from the analysis. The period depends only on the length of the pendulum and the acceleration due to gravity.

💡Amplitude

Amplitude in the context of the pendulum refers to the maximum displacement from the pendulum's equilibrium position. The script mentions that the period of the pendulum's motion does not depend on the amplitude, which is a significant result in the study of simple harmonic motion. This means that regardless of how far the pendulum is displaced, the time taken for one complete oscillation remains constant.

💡Taylor Series

A Taylor series is a mathematical representation of a function as an infinite sum of terms calculated from the values of its derivatives at a single point. In the script, the Taylor series is mentioned in the context of the small angle approximation, where the sine function is expanded as 'theta - theta cubed over three factorial + ...'. This series is used to justify the approximation 'sine theta is approximately theta' for small angles, which simplifies the analysis of the pendulum's motion.

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