How to answer Oscillations Questions in A Level Phyics (Simple Harmonic Motion, Resonance)

Summary

This video tutorial focuses on tackling oscillation and simple harmonic motion questions for A-level physics exams. The instructor guides viewers through past exam questions, emphasizing the importance of understanding key equations and interpreting graphs. The video covers topics like pendulum oscillations, deriving angular frequency, and resonance, providing step-by-step explanations to solve problems efficiently. It also highlights the significance of memorizing fundamental formulas and offers tips for excelling in exam questions related to oscillations.

Takeaways

• 📚 The video focuses on answering oscillation and simple harmonic motion questions for A-level physics, aiming to simplify the complex topic by working through past exam questions.
• 🔍 The presenter uses a 2022 meijing paper to illustrate a typical simple harmonic motion question involving a pendulum, emphasizing the importance of understanding the relationship between acceleration and displacement.
• 📉 The video explains how a graph with a negative gradient indicates that the acceleration is in the opposite direction to the displacement, a key characteristic of simple harmonic motion.
• 🔢 The formula for acceleration in simple harmonic motion, a = -ω²x, is derived and used to calculate the angular frequency (ω) from the graph's gradient.
• 🔄 The video discusses how the angular frequency (ω) relates to the length of a pendulum and demonstrates how to calculate a constant (K) using the previously determined angular frequency.
• 🌀 The concept of resonance is introduced, explaining that it occurs when an object's driving frequency matches its natural frequency, leading to increased amplitude.
• 📈 The script guides viewers on how to calculate the period of oscillation using the angular frequency and provides a method to label axes correctly with time and displacement scales.
• 🕒 The video demonstrates calculating phase difference by comparing the time at which each pendulum crosses the equilibrium position, emphasizing the importance of understanding periodicity.
• 📝 The presenter advises memorizing key equations and concepts to efficiently answer questions, such as those involving resonance and simple harmonic motion.
• 🎓 The video concludes by encouraging practice and memorization of key formulas and concepts for success in answering A-level physics questions on oscillations.

Q & A

What is the main topic discussed in the video?

The video discusses how to answer questions on oscillation and simple harmonic motion in A-level physics, focusing on understanding and applying various equations and formulas related to these concepts.

Why is simple harmonic motion considered a daunting topic for students?

Simple harmonic motion is considered daunting because it involves many equations and formulas, and there is a substantial amount of content in the chapter that can be overwhelming to learn and apply.

What is the significance of the negative gradient in the graph shown in the video?

The negative gradient in the graph indicates that the acceleration is in the opposite direction to the displacement, which is a key characteristic of simple harmonic motion.

How does the video suggest determining the angular frequency of an oscillation?

The video suggests using the given formula a = -Omega^2 * x and the information from the graph to determine the angular frequency (Omega) of an oscillation.

What is the relationship between angular frequency and the length of a pendulum as explained in the video?

The video explains that the angular frequency (Omega) is related to the length (L) of the pendulum by a constant (K), and this relationship can be used to determine the value of K using the previously calculated angular frequency.

How does the video explain the concept of resonance?

Resonance is explained as the phenomenon where an object reaches its maximum amplitude when the driving frequency equals the natural frequency of the object.

What is the importance of understanding the units of angular frequency as discussed in the video?

Understanding that angular frequency is a ratio and does not have units (it is dimensionless) is important for correctly interpreting and calculating the values in problems involving simple harmonic motion.

How does the video suggest calculating the period of oscillation given a displacement-time graph?

The video suggests using the formula for displacement in simple harmonic motion, x = x_0 * sin(Omega * t), to find the angular frequency (Omega) and then calculate the period (T) as T = 2 * pi / Omega.

What is the qualitative effect on the amplitude of oscillations when the length of the string in a pendulum is increased, as discussed in the video?

When the length of the string in a pendulum is increased while maintaining the same total energy, the amplitude of oscillations increases because the angular frequency decreases, requiring a larger displacement to maintain the same energy.

How does the video approach the calculation of phase difference between two oscillating pendulums?

The video approaches the calculation of phase difference by determining the time difference at which the pendulums cross the equilibrium position and then converting this time difference into a phase angle in radians.

Outlines

00:00

📚 Introduction to Oscillation and Simple Harmonic Motion

The script begins with an introduction to answering questions on oscillation and simple harmonic motion in A-level physics. It acknowledges the complexity of the topic due to the multitude of equations and formulas involved. The speaker suggests that practicing past papers is an effective way to streamline the learning process. The focus then shifts to a specific question from the 2022 Meijing paper, which involves a pendulum with a small metal sphere (bob) attached to a string. The pendulum's oscillation and its relationship with the string's length and the bob's displacement from the equilibrium position are discussed. The speaker emphasizes the importance of understanding the graph showing the acceleration's variation with displacement, highlighting the negative gradient and the linear relationship indicative of simple harmonic motion.

05:02

🔍 Analyzing Simple Harmonic Motion with Graphs

This section delves deeper into analyzing the pendulum's motion using a graph. The speaker explains how the graph's negative gradient and linearity demonstrate the direct proportionality between acceleration and displacement, which are key characteristics of simple harmonic motion. The discussion then transitions into a mathematical approach, using the formula for angular frequency (Omega) in simple harmonic motion. The speaker rearranges the formula to solve for Omega, using the given data from the question. The result is an angular frequency of 2.83 radians per second. The speaker also touches on the concept of radians as a ratio, clarifying that they are unitless and used to describe angular relationships.

10:03

🔗 Relating Angular Frequency to Pendulum Length

The script continues with a question that connects the angular frequency (Omega) to the length (L) of the pendulum. The speaker is tasked with determining a constant (K) from the relationship between these two variables. Using the previously calculated angular frequency and the given length of the pendulum, the speaker solves for K. The importance of unit consistency is highlighted, with a detailed explanation of why radians are dimensionless. The discussion then moves to the qualitative effect of increasing the pendulum's length on the oscillation's amplitude, explaining that to maintain constant total energy, a decrease in angular frequency must be compensated by an increase in amplitude.

15:05

🌊 Exploring Resonance in Oscillating Systems

The final paragraph shifts the focus to resonance, a phenomenon where the amplitude of an oscillating system increases significantly when driven at its natural frequency. The speaker defines resonance and connects it to the concept of maximum amplitude occurring when the driving frequency matches the natural frequency. The script presents a scenario involving two pendulums with the same natural frequency but different masses, illustrating how the heavier pendulum has a larger amplitude of oscillation. The speaker then calculates the period of oscillation for the light pendulum using the given displacement-time relationship. The axes of a graph are labeled accordingly, and the amplitude is determined from the maximum value of the sine function. Finally, the phase difference between the oscillations of the light and heavy pendulums is calculated, demonstrating the application of concepts from the oscillations chapter.

Keywords

💡Oscillation

Oscillation refers to the repetitive variation, typically in time, of some measure about a central value or between two or more alternative conditions. In the context of the video, oscillation is central to understanding the behavior of pendulums and other physical systems. The script discusses how pendulums oscillate with small movements about their equilibrium position, illustrating the concept with a bob attached to a string.

💡Simple Harmonic Motion (SHM)

Simple Harmonic Motion is a type of periodic motion where the restoring force acting on the system is directly proportional to the displacement and always directed towards the equilibrium position. The video emphasizes SHM as a prerequisite for the pendulum's motion, highlighting the direct proportionality between acceleration and displacement, which is a key characteristic of SHM.

💡Equilibrium Position

The equilibrium position is the central position about which an oscillating system moves. In the video, the equilibrium position is where the pendulum's bob would be if no external forces were acting on it. The script mentions that the pendulum oscillates about this equilibrium position, which is a fundamental concept in understanding oscillations.

💡Angular Frequency (Omega)

Angular frequency, denoted by Omega, is a measure of how fast an object rotates or oscillates, expressed in radians per second. The video script uses the concept to calculate the rate at which the pendulum's bob oscillates. It is derived from the given acceleration and displacement relationship, showcasing its importance in analyzing the motion of the pendulum.

💡Restoring Force

The restoring force is the force that brings an object back to its equilibrium position when it is displaced. In the context of the video, the restoring force is what causes the pendulum's bob to oscillate. The script explains that this force increases as the displacement from the equilibrium position increases, which is a characteristic of SHM.

💡Displacement

Displacement in the video refers to the change in position of the pendulum's bob from its equilibrium position. It is a vector quantity that is crucial for understanding the motion of the pendulum. The script describes how the acceleration of the bob varies with its displacement, which is depicted in a graph, indicating its importance in analyzing the pendulum's motion.

💡Resonance

Resonance is a phenomenon where the amplitude of an oscillating system increases when it is driven at or near its natural frequency. The video script discusses resonance in the context of two pendulums with the same natural frequency, explaining how the amplitude of oscillation can increase significantly when the driving frequency matches the natural frequency.

💡Natural Frequency

Natural frequency is the frequency at which a system oscillates when not driven by an external force. In the video, the natural frequency is mentioned in relation to resonance, where the heavy and light pendulums have the same natural frequency. This concept is crucial for understanding how external forces can affect the oscillation of a system.

💡Phase Difference

Phase difference refers to the difference in the phase angle between two waves or oscillating systems. In the video, the script calculates the phase difference between the oscillations of a light and heavy pendulum. Understanding phase difference is important for analyzing how two systems may be out of sync with each other.

💡Amplitude

Amplitude is the maximum extent of a displacement from the equilibrium position in an oscillating system. The video script discusses how the amplitude of the pendulum's oscillations can change if the length of the string is altered while maintaining the total energy of the oscillations. Amplitude is a key parameter in describing the motion of oscillating systems.

Highlights

Introduction to answering oscillation and simple harmonic motion questions in A-level physics.

Emphasis on the importance of past papers for streamlining learning and understanding common question types.

Analysis of a pendulum example from the 2022 Meijing paper to illustrate typical simple harmonic motion questions.

Explanation of the relationship between pendulum length, equilibrium position, and maximum displacement.

Discussion on how a linear graph with a negative gradient indicates simple harmonic motion.

The significance of the negative gradient in showing the opposition between displacement and acceleration.

Advice on memorizing key concepts to save time during exams.

Derivation of angular frequency using the formula a = -omega^2 x and its rearrangement for solving problems.

Clarification on the units of angular frequency and the concept of radians as a ratio.

Use of the derived angular frequency to determine a constant K in the equation relating to pendulum length.

Explanation of the qualitative effect of changing the length of a pendulum on its amplitude and angular frequency.

Introduction to resonance and its significance in oscillation questions.

Definition of resonance and its relation to maximum amplitude and driving frequency.

Analysis of a diagram showing the oscillations of a heavy and a light pendulum with the same natural frequency.

Calculation of the period of oscillation using the given displacement-time relationship.

Guidance on labeling axes with correct scales and additional working for pendulum oscillation graphs.

Determination of the phase difference between the oscillations of the light and heavy pendulums.

Advice on practicing and memorizing key equations and concepts for oscillation and resonance questions.

Conclusion and encouragement for viewers to explore more physics A-level videos for further understanding.

Transcripts

video I want to talk about how to answer

questions on oscillation and simple

harmonic motion in a levels and it's a

very daunting topic to start answering

questions about because there are very

many equations and formulas and there's

just a lot of substance in that chapter

so hopefully by doing past papers it

really streamlines the amount of

information that you should be learning

so we're going to work through a bunch

of different questions from various

papers all pretty recent to look at what

they're actually going to ask of you

so let's start with this one this one is

in the meijing paper in 2022 It's Paper

four one and I think it's a very typical

example of what a simple harmonic motion

or an oscillation question looks like

so over here it says that a pendulum

consists of a bob which is a small metal

sphere attached to that piece the end of

a piece of a string and then the other

end is connected to a fixed spot and the

Bob oscillates with small oscillations

about its equilibrium position so this

is exactly what they're telling us and

it's fixed over here and it's

essentially just telling us it's going

to oscillate over here what we should be

paying attention to is the length so L

of the string is given over here and we

know that equilibrium is over here

maximum displacement is given by X so

you should all note that and then they

say that the length of the pendulum

measured from the fixed point to the

center of the Bob is 1.24 meters and the

acceleration a of the Bob varies with

its displacement X from the equilibrium

position as shown in figure 8.2 so we

see that the acceleration to the

distance the maximum displacement is

constant for one because it has straight

line so this is a linear graph and it's

also it has a negative gradient so this

is really important and it comes up over

and over again because in the very next

question they're going to ask you state

how figure 0.4.2 shows that the motion

of the pendulum is simple harmonic and

we know that a simple harmonic motion

has an acceleration that is directly

proportional to the displacement of the

object from the equilibrium position and

we also know that the force the

restoring force that pushes the object

back into this equilibrium position it

gets bigger the bigger the X becomes so

we know that these two are the

prerequisites of simple harmonic motion

and so we all we need to do is just use

certain keywords to analyze the graph

and that would be this

so the negative gradient shows that the

displacement is in the opposite

direction to the acceleration and this

is very

self-explanatory because we know that

the positive of something would have the

negative of other things so to really

exemplify what I mean by that if we have

a positive of X we would have a negative

of a which means that they are always

going to be in opposite directions to

each other so that's the first part

and this is the second part the straight

line shows that the acceleration is

directly proportional to the

displacement so previously I talked

about how the straight line means that

the curve the graph is linear so if

there is a linear relationship then

there is a direct proportionality

between the two constants so the higher

something is the higher the other

whatever is so that is what you should

be commenting on these two things

they're really straightforward and this

is a question that comes up super often

so I would really recommend just knowing

this by heart to save time now we have

to get into some math so they tell us to

use figure Point 4.2 to determine the

angular frequency which is given by

Omega of the oscillation so over here

you have to memorize a formula for a

simple harmonic motion and that formula

is a equals

negative Omega square and if you want to

derive this do check out this one video

where I talk about all of the equations

of oscillation simple harmonic motion

and how to derive them and it is

actually also given to you over here

so you have this equation given to you

which is very nice and it's also worth

taking a look at what is given to you in

terms of simple harmonic motion these

very like primary equations are given to

you so you wouldn't have to waste a lot

of time trying to memorize all of them

but we have this very simple equation

that you can get from the beginning of

the paper and all we have to do now is

to basically just substitute so I can do

that here

I can first of all

rearrange this

so the answer is 2.83 so it is 2

.83 radians per second so it would spin

by this much

that would be the angular frequency

so now we've done that we can go on to

this part so they tell us that the

angular frequency Omega is related to

the length L of the pendulum by this

equation

where K is a constant and they tell us

to use your answer in B1 to determine K

and give a unit with your answer so we

want to be very particular with the

units as well so it's it makes a lot of

sense to write down everything that

we're doing so we know that 2.83

radians

per second equals to K over

the length which is you know we know

that L is 1.24 from the data given to us

beforehand so 1.24 meters

so before we get any further on with

this it's time to think about the radian

what does a radian actually mean the

radian is a ratio

between so if you had a sector of a

circle it's the ratio between this and

the radius and this has a unit of meters

this also has a unit of meters so it's a

ratio of these two which means that

there are no units because the meters

would cancel out so you can also just

say that if you purely want to talk

about the units it's 2.83

second to the power of negative one

so getting the value in and of itself is

not hard what is tricky is remembering

that radians don't actually have units

they're purely ratio and so this would

be the ultimate unit that you get so I'd

write down your 9.7

meters per second squared

now we have the final part of this

question so while the pendulum is

oscillating the length of the string is

increased in such a way that the total

energy of the oscillations remains

contact so Justin explain the

qualitative effect of this change on the

amplitude of the oscillations

so what is the key word here is that the

total energy of the oscillation is

remains contact I want you to imagine in

oscillating pendulum that's going back

and forth

is lengthened

like this

at the end of the day the pendulum and

the Bob will have to be moving much

greater distances so if it continues to

do so at the same speed the angular

frequency

which is the frequency of how much it is

displaced angularly this would just

decrease If This Were to stay rather

similar so we know that there will be a

decreased Omega decreased angular

frequency now we want to make sure that

the

um

energy Remains the Same and the way that

this would happen is if the amplitude

increased so if the Omega decreases

um and the amplitude stays the same this

means that the longer

pendulum is simply going to

oscillate

in the same maximum displacement back

and fro but that's not really what

happens it's more likely to extend all

the way here and move here to here so

this is why in order to make sure that

the energy stays the same we need to

decrease the angular frequency to offset

that energy change and increase that

amplitude because we can see that the x

is going to become the maximum

displacement is going to become even

bigger than it was before so here is the

answer

[Laughter]

so this is my answer here so we can just

rub this off and this is essentially

what will be happening if you pictured

the pendulum and what would happen

um purely

intuitively we know that if the maximum

displacement was this and you lengthen

the pendulum it's not going to start

oscillating towards this much it's going

to start oscillating like this so it's

going to be slower in terms of angular

frequency in order to increase that

amount of amplitude I hope that makes

sense so this is like the first example

we have of a question it's really

typical the really the only equation

that we had to use was this equation and

this equation is very common it comes up

all the time and you can easily find it

in the front of the paper a big part of

the oscillations chapter is not simple

harmonics but resonance and this is a

question regarding resonance so it's

very likely that there will be one

question in your paper that is going to

be about oscillations and it's going to

be either on simple harmonic motion or

on resonance or it's going to be on both

of them and somehow they're going to

have like one part you know regarding

simple harmonic motion and then the

latter part regarding the resonance or

whatever so this question is mainly

about resonance so we can take a look at

um how to answer this question so first

of all it's really simple and easy they

ask us State what is meant by the

resonance and we know what is resonance

imagine something here it's already

moving by itself and then we put a hand

here at the same frequency of its

movement and we add up add on to it and

the amplitude will increase a lot so how

you would write this is this

[Applause]

so that's definition you basically have

the object being at a maximum amplitude

when the driving frequency equals the

natural frequency of the object the

keywords that you should look out for

here is you should always remember to

talk about the maximum amplitude

and then you should also include the

world driving frequency and the natural

frequency so those are the keywords that

you have to include in in the definition

here so let's continue onwards

so it says that the figure 4.1 shows a

heavy pendulum and a light pendulum and

they're both suspended from the same

piece of the string and this string is

secured at each end and fixed two points

so we have this sort of like complex

system where there are two

pendulums so there's one heavy one and a

light one now they have the same natural

frequency the heavy pendulum is set to

oscillating perpendicular to the plane

of the diagram which means it's going to

come out and go back in

um because the plane of the diagram is

what we're seeing right now on the paper

or on the screen

so this is essentially the variation

that the graph shows us of time T of the

displacements of the two pendulums for

three oscillations so we have the heavy

pendulum

it's oscillating like this so there are

three oscillations and then we have the

light pendulum it's oscillating like

this and there are three

oscillations so it looks you know like

they're

you know not in face but they actually

have the same frequency because of the

fact that they exactly contain

three oscillations so that's the first

thing that we should realize here and we

also see that the heavy pendulum is

moving with a much higher oscillation

amplitude than the light one but other

than the amplitudes

they are basically have the same

frequency so now the question finally

tells us the variation with t of the

displacement acts of the light pendulum

is given by this which is in centimeters

and T is in second so they ask us to

calculate the period T of the

oscillation

so this is essentially what of the

equation is kind of derived out of so

the velocity of particle in simple

harmonic motion and we have the

variation with t of the displacement of

X in the light pendulum which means that

essentially the only thing that we

should care about is the Omega here

this is telling us that 5.0 pi equals

the Omega so Omega equals 5.0 pi and we

know that the angular frequency is

actually the amount of time that it

takes for one full oscillation

so it's basically telling us the amount

of

the angle value in pi

um

that is basically

moved through per unit time so that's

actually two pi over period T because

whatever T is it's going to move through

2 pi in that time it's going to do a

full oscillation during that time so we

just have to do some substitution and

that's all we need so we know that 2 pi

over T equals 5.0 Pi so we know that t

equals 2 pi over 5.0 Pi which gives us

0.4 seconds so that's our answer right

here

and now they tell us to label both of

the axes with the correct scales and use

the space below for any additional

working so now that we know the the time

this becomes quite easy to do this so we

know that it makes one oscillation in

0.4 seconds so this is just very easy

for us 0.4

this is another oscillation that's 0.8

and then that would be

1.2 and you know you can also fill in

the gaps in between 0.2 0.6

1.0 stuff like that now we should talk

about the amplitude so if you go back

here

we see this right here

this is basically showing us the maximum

displacement of the

um

the particle X because

when for instance there is maximum

um amplit and when there is maximum

displacement that means that the

sine over here is going to be at a

maximum value and it's going to equal 1

right so that means maximum displacement

has to be at this time 0.25 that's the

highest it's going to get so we know

that that is the case for the light

pendulum so this is 0.25 and that makes

life a lot easier for us this is 0.5

0.75 and one and then you can also put

in the negative

value if you want to and so that's how

you would fill in everything it's pretty

straightforward if you wanted to you

could also make it as detailed as

possible just in case there are marks

given for adding extra values whenever

you can so I would also suggest just if

you have enough time

filling out all of the values

in detail

and now we have the final one the final

question asks us to determine the

magnitude of the phase difference

between the oscillations of the light

and heavy pendulums and we have to give

a unit with our answer so the phase

difference is quite easy to find we know

that over here

we can take a close look at this and we

know that this guy

goes back up and

hits the x-axis at 0.4 but this guy goes

back up and hits the x-axis at 0.3 so

the ultimate time difference that we

have here is 0.1 seconds and we know

that the uh the periods of both of these

oscillations are the same the periods

are both 0.4 so if you want to find the

phase difference all you have to do is

to find the fraction of the difference

of time

as a fraction of the total period so

what I mean by that is you should just

write 0.1 out of 0.4 and it times this

fraction by 2 pi and this if you do that

you're going to get half pi radians

and that is the answer that's a look at

how to calculate these resonance

questions what's really important here

in the oscillations chapter is to be

able to just use the equations that are

given to you in the beginning of the

paper very well so this is the velocity

of a particle a simple harmonic motion

but we didn't use it as velocity we

actually used it for displacement so x

equals x o sine

Omega T and it's in the same format and

if you want to find the acceleration you

would also find that it's in the same

format

um we have used this one for simple

harmonic motion and we can use this if

the situation arises for it as well it's

good to be able to tie in various

formulas that are often used to these

things and you would find that most of

the formulas that are required of you

you can actually kind of derive from

what you have over here instead of

having to memorize everything and

obviously they're really important very

common questions that occur in these

chapters such as Define what resonance

is and what is simple harmonic motion

what are the two things that are

required for that motion so these are

kind of the skeleton of

um what really constitutes a question in

oscillation resonance I think they're

very typical examples of what you might

get and it's a good idea to practice

things like this and hopefully be able

to memorize how to answer a lot of the

wordy parts so yeah so I do hope that

this video kind of gave you a good idea

of how to go about in these structured

questions about oscillations and I hope

it was helpful ultimately if you want

more videos such as this on physics and

a levels then do check out the other

videos I have on my channel thank you so

much for watching