AP Physics Workbook 6.C Equations of Motion for Simple Harmonic Motion

Mr.S ClassRoom
14 Apr 202015:59
EducationalLearning
32 Likes 10 Comments

TLDRThe video script discusses the principles of simple harmonic motion, focusing on a cart-spring system. It explains concepts such as frequency, wavelength, period, amplitude, and their relationship with the system's parameters. The impact of varying mass and spring constant on the period of oscillation is also explored. The script uses graphical data and equations, such as x = A cos(Ο‰t) and x = A sin(Ο‰t), to describe the system's behavior, highlighting how different starting conditions can lead to different representations of the motion.

Takeaways
  • πŸ“š The topic is AP Physics, focusing on simple harmonic motion and equations of motion.
  • 🏎️ A cart of mass M on a smooth surface is attached to an ideal spring and undergoes simple harmonic motion.
  • πŸ“ˆ The motion detector collects data to create a graph of position vs. time for analysis.
  • 🌊 Frequency is the number of cycles per second, measured in Hertz (Hz), and wavelength is the distance from peak to peak.
  • πŸ”„ One period (T) is the time for one complete oscillation, and is related to frequency by the formula T = 1/f or f = 1/T.
  • πŸ“ Amplitude is the maximum displacement from equilibrium, representing the peak and trough of the oscillation.
  • πŸ“ˆ The graph shows that the cart's oscillation completes a full cycle in 4 seconds, resulting in a frequency of 0.25 Hz.
  • πŸ€” The angular velocity (Ξ©) is calculated using the formula Ξ© = 2Ο€/T, which simplifies to Ξ© = Ο€/2 for this scenario.
  • πŸš€ The maximum positive and negative accelerations occur at different times during the oscillation, pointing towards the center of the oscillation path.
  • πŸ“ˆ The position of the cart is described by a cosine wave, with the equation x = A*cos(Ο‰t), where A is the amplitude and Ο‰ is the angular frequency.
  • πŸ”§ The period T is influenced by the mass (M) and the spring constant (K), following the formula T = 2Ο€βˆš(M/K), indicating that a larger mass results in a longer period and a stiffer spring results in a shorter period.
Q & A
  • What is the definition of frequency in the context of the script?

    -In the context of the script, frequency is defined as the number of cycles or the number of wavelengths passing through a position in one second. The unit of frequency is Hertz, which means one cycle per second.

  • How is the wavelength described in the script?

    -The wavelength in the script is described as the distance measured from peak to peak, which represents one complete wavelength.

  • What is the relationship between period and frequency?

    -The relationship between period and frequency is that they are inversely related. The period (T) is defined as the time for one complete oscillation or cycle, and it can be calculated as 2Ο€/Ξ© (which is equivalent to 1/f), where f is the frequency.

  • What is simple harmonic motion?

    -Simple harmonic motion is a type of periodic motion where an object moves back and forth around an equilibrium position in a repetitive and regular manner. The motion is characterized by the object being fully compressed and stretched, reaching maximum amplitude at these points.

  • How does the angular velocity (Ο‰) of the cart in the script relate to the period (T)?

    -The angular velocity (Ο‰) is related to the period (T) by the formula Ο‰ = 2Ο€/T. The script provides an example where the period T is 4 seconds, so Ο‰ would be Ο€/2 rad/s.

  • What happens to the period (T) when the mass (M) of the cart increases?

    -When the mass (M) of the cart increases, the period (T) also increases. This is because a larger mass has more inertia and thus requires more time to complete one oscillation, leading to a longer period. The period is given by the formula T = 2Ο€βˆš(M/K), where K is the spring constant.

  • How does the spring constant (K) affect the period (T)?

    -The spring constant (K) has an inverse relationship with the period (T). A stiffer spring (higher K value) results in a shorter period because the spring exerts a greater restoring force, allowing the cart to oscillate more quickly. This relationship is reflected in the formula T = 2Ο€βˆš(M/K).

  • What is the significance of the starting value on the graph?

    -The starting value on the graph indicates the initial position of the cart. In the script, one group starts with the cart fully compressed (a starting value of 4), while the other group starts with the cart at equilibrium (a starting value of 0). This difference affects the form of the equation used to describe the motion, with the latter being described by a sine wave equation instead of a cosine wave equation.

  • How does the position of the cart over time change when the mass is quadrupled?

    -When the mass is quadrupled, the period of the cart's oscillation increases, resulting in a slower motion. The graph would start at the same position but take longer to reach its peak and trough positions. The motion still exhibits simple harmonic motion, but the cycles occur less frequently due to the increased mass.

  • What is the difference between a cosine wave and a sine wave as described in the script?

    -In the context of the script, a cosine wave describes the motion of the cart when the sensor starts recording when the cart is fully compressed, while a sine wave describes the motion when the sensor starts at equilibrium. Both waves represent simple harmonic motion, but they differ in their phase, with the cosine wave starting at the maximum displacement and the sine wave starting at the equilibrium position.

  • How is the position of the cart described mathematically in the script?

    -The position of the cart is described mathematically using the equation of a cosine wave when the sensor starts at the fully compressed position (x = A cos(Ο‰t + Ο†)), and a sine wave when the sensor starts at equilibrium (x = A sin(Ο‰t + Ο†)), where A is the amplitude, Ο‰ is the angular velocity, t is time, and Ο† is the phase constant.

Outlines
00:00
πŸ“š Introduction to Simple Harmonic Motion

This paragraph introduces the concept of simple harmonic motion in the context of a cart attached to a spring. It explains the scenario where a cart of mass M is displaced and then released, leading to oscillatory motion around an equilibrium position. Key terms such as frequency, wavelength, amplitude, and period are defined, with an emphasis on their relevance to understanding the motion. The paragraph also touches on the idea that these oscillations are repetitive and periodic, which is a hallmark of simple harmonic motion. The goal is to provide a foundational understanding of the vocabulary and concepts necessary to analyze and work with problems involving simple harmonic motion.

05:02
πŸ“ˆ Analysis of Velocity and Acceleration

In this paragraph, the focus shifts to analyzing the velocity and acceleration of the cart during its oscillation. It describes the points of maximum positive and negative velocity, as well as when the velocity is zero. The discussion also covers the direction of acceleration, highlighting that it is always directed towards the equilibrium position, indicating a restoring force. The paragraph uses the graph of the cart's motion to illustrate these points, providing a clear visual representation of the concepts. This analysis is crucial for understanding the dynamics of the system and how it behaves during simple harmonic motion.

10:07
πŸ“Š Graph Comparison and Equation Derivation

This paragraph compares two different sets of data collected from the same oscillating system but with different starting conditions. It explains how the starting position affects the form of the graph, transitioning from a cosine wave to a sine wave. The paragraph then derives the equations of motion for both scenarios, emphasizing the mathematical representation of the cart's behavior. It also discusses the impact of changing the mass on the period of oscillation, providing a deeper understanding of how the physical properties of the system influence its motion. This section is essential for relating the graphical analysis to the underlying mathematical equations and for understanding how system parameters affect the motion.

15:13
πŸ”§ Impact of Mass and Spring Constant on Period

The final paragraph delves into the effects of mass and spring constant on the period of the oscillation. It explains how increasing the mass leads to an increase in the period, while increasing the spring constant results in a decrease in the period. This relationship is crucial for predicting and tuning the behavior of a system undergoing simple harmonic motion. The paragraph reinforces the understanding of how the physical characteristics of the system, such as mass and spring stiffness, play a role in determining the nature of its oscillations.

Mindmap
Sine Wave
Cosine Wave
Maximum Negative Acceleration
Maximum Positive Acceleration
Velocity Equals Zero
Maximum Negative Velocity
Maximum Positive Velocity
Understanding Implications
Different Starting Points
Spring Constant Impact
Mass Impact
Period Equation
Equation Representation
Time Points
Position vs. Time Graph
Simple Harmonic Motion
Period
Amplitude
Wavelength
Frequency
Comparative Analysis
Effects of Mass and Spring Constant
Graph Analysis
Basic Concepts
AP Physics - Simple Harmonic Motion
Alert
Keywords
πŸ’‘Simple Harmonic Motion
Simple Harmonic Motion (SHM) is a type of periodic motion where an object oscillates repeatedly in a straight line around an equilibrium position. In the video, this concept is exemplified by the cart attached to a spring, which oscillates back and forth around its equilibrium point. The motion is characterized by its regularity and repetitive nature, with the cart moving in a predictable pattern over time.
πŸ’‘Frequency
Frequency is the number of complete cycles or oscillations that occur in one second. It is a measure of how often an event repeats and is typically measured in Hertz (Hz). In the context of the video, the frequency is used to describe the rate at which the cart oscillates around the equilibrium position.
πŸ’‘Wavelength
Wavelength is the distance over which one complete cycle occurs, typically measured from peak to peak in the context of waves. In the video, the wavelength is related to the cart's motion as it describes the distance between two successive points in the same phase of the oscillation, such as two consecutive maximum displacements of the cart.
πŸ’‘Amplitude
Amplitude refers to the maximum displacement of an oscillating object from its equilibrium position. It quantifies the extent of the motion and is a measure of the energy in the system. In the video, the amplitude is the distance the cart moves away from its equilibrium position, either to the maximum height or through the maximum stretch of the spring.
πŸ’‘Period
The period is the time taken for one complete cycle of the oscillation to occur. It is the duration from the start of one oscillation to the start of the next identical point in the cycle. In the video, the period is determined by how long it takes for the cart to complete one full back-and-forth motion from the equilibrium position and return to it.
πŸ’‘Angular Velocity
Angular velocity, denoted by the Greek letter Omega (Ο‰), is the rate at which an object rotates or revolves around an axis. It measures the angle covered per unit time and is typically given in radians per second. In the context of the video, angular velocity is related to the cart's oscillation around the equilibrium position, describing how quickly the cart moves through angles as it oscillates.
πŸ’‘Equilibrium Position
The equilibrium position is the central or balanced position around which an object oscillates in simple harmonic motion. It is the point where the restoring force is zero and the object is in a state of no net force. In the video, the equilibrium position is the central point on the graph where the cart would be if no external forces were acting upon it.
πŸ’‘Cosine Wave
A cosine wave is a type of periodic wave that uses the cosine function to describe its oscillation. It is a mathematical representation of simple harmonic motion and is characterized by its smooth, repeating pattern. In the video, the position of the cart is described by a cosine wave equation, which models its back-and-forth motion around the equilibrium position.
πŸ’‘Sine Wave
A sine wave is similar to a cosine wave but is phase-shifted by 90 degrees. It is another way to mathematically describe periodic motion such as the oscillation of the cart in the video. A sine wave uses the sine function to represent the pattern of motion over time.
πŸ’‘Mass
In the context of the video, mass refers to the amount of matter in an object, which affects its inertia or resistance to changes in motion. The mass of the cart in the simple harmonic motion scenario influences the period of its oscillation, with greater mass leading to a longer period and slower oscillations.
πŸ’‘Spring Constant
The spring constant, often denoted by 'k', is a measure of the stiffness of a spring. It describes the amount of force required to stretch or compress the spring by a certain distance. In the video, the spring constant is a key factor in determining the period of the cart's oscillation, with a stiffer spring (higher k value) resulting in a shorter period and faster oscillations.
Highlights

The scenario involves a cart of mass M attached to an ideal spring, oscillating around an equilibrium position.

The cart's displacement to the right is denoted as distance Delta X from equilibrium.

A motion detector collects data to create a graph of position versus time.

Frequency is defined as the number of cycles or wavelength passing per second, with units in Hertz.

Wavelength is measured from peak to peak, representing one complete cycle.

One period T is the time for one complete oscillation or one cycle from crest to crest and back to equilibrium.

Amplitude is the maximum displacement from equilibrium to the highest or lowest point.

Simple harmonic motion occurs when a force causes repetitive, periodic motion.

The period T is the time taken to complete one full cycle, and it's inversely related to frequency.

Angular velocity, denoted by Omega, is calculated using the formula Omega = 2PI/T.

The maximum positive and negative velocities occur at different time intervals during the oscillation cycle.

The position of the cart is described by a cosine wave equation, x = A cos(Ο‰t + Ο†), where A is the amplitude, Ο‰ is the angular frequency, and Ο† is the phase constant.

Different starting conditions, such as beginning at full compression or at equilibrium, result in different types of wave equations (cosine or sine).

The period T is affected by the mass of the cart and the stiffness of the spring, with the formula T = 2Ο€βˆš(M/K).

An increase in mass M results in an increase in the period T, leading to slower oscillations.

A stiffer spring, represented by an increase in K, results in a shorter period T and quicker oscillations.

The theoretical understanding of simple harmonic motion and its equations are essential for analyzing physical systems.

Transcripts
00:00

all right welcome this is the AP Physics

00:03

workbook solution here we have unit six

00:06

in part monic motion the section is six

00:08

point C equations of motion for simple

00:11

harmonic here here's the scenario

00:15

a cart of mass M resting on a smooth

00:18

surface is attached to an ideal spring

00:20

the cart is displacement to the right as

00:22

the distance Delta X from equilibrium

00:25

and release while the car oscillates

00:26

around the equilibrium position a motion

00:29

detector collects data and to make the

00:31

following graph of position as a

00:33

function of time before we read this I

00:35

would like to give you some notes all

00:38

right so we understand the right vocab

00:42

that we're using the frequency is the

00:44

number of cycles or the number of

00:46

wavelength passing through a position in

00:48

one second the units of frequency is

00:51

considered car Hertz which means one

00:54

second the wavelength is measured from

00:56

peak to peak

00:57

so from here peak to peak or any okay so

01:03

here it would be one complete it would

01:06

be consider one wavelength here right so

01:09

Springs are like waves and circles okay

01:11

here's one period a period T is the time

01:17

for one revolution or in case in this

01:21

case spring the time for one complete

01:24

oscillation or one crest and one trough

01:26

one cress one trough and it comes back

01:29

here to equilibrium okay oscillations

01:31

can be also we call vibrations and

01:34

cycles this is the amplitude from

01:37

equilibrium into the maximum height is

01:41

called the amplitude okay as you go to

01:45

the right it's called the equilibrium

01:47

it's in the equilibrium time period is

01:52

defined as 2 PI over Omega or 1 over f

01:55

which is considered the frequency notice

01:58

time and frequencies are inverse of each

02:00

other okay a more interesting situation

02:04

develops when we exert the same force on

02:06

the system it repeats in in repetitive

02:09

periodic motion this is the simple

02:11

harmonic motion

02:13

you should see that it is fully

02:15

compressed aptitude is its maximum

02:18

height here at the amplitude here it's

02:22

fully stretch which is at its trough

02:25

here do you see how it comes back to its

02:27

original position it completes its full

02:30

cycle here that's called T the time it

02:33

takes to complete one full cycle a

02:35

stands for the amplitude and for the

02:37

frequency and T is for the period good

02:40

you're gonna need these ideas to help

02:43

you do this if you would like a more

02:44

detailed explanation on the notes please

02:46

watch the simple harmonic lecture notes

02:49

alright so let's see how long it take

02:53

for T for the period so how long did it

02:58

take for this to complete one full cycle

03:01

so it starts from here and it goes all

03:05

the way until it reaches this so it goes

03:09

from zero all the way to four so we

03:11

could write here four seconds okay

03:14

because this is its complete cycle now F

03:19

is considered its frequency so you can

03:22

just write it right here T is equal to 1

03:25

over F and we know that here it is 4 so

03:31

you can write you can make the

03:33

substitution here T equals to 4 multiply

03:42

frequency is equal to 1/4 you can also

03:45

write that as 0.25 the units for

03:50

frequency is considered like we said

03:53

here is call hertz which is per second

03:57

so the frequency here is 0.25 Hertz so

04:04

point two five of the wave occurred in

04:09

one second

04:10

that's what this means now W which is

04:14

which is its angular velocity okay how

04:20

do we get its angular velocity we can

04:22

use this formula right here

04:26

is equal to 2 PI over Omega now we can

04:31

solve for W or Omega okay so T is for

04:38

two PI over W cross multiply you get W

04:43

Omega is equal to two PI over four which

04:46

can be simplified to just PI over two so

04:51

you could say here this is PI over two

04:54

the units for this is rad over seconds

04:59

all right here's the math if you would

05:01

like to see it okay do you need to

05:06

memorize this formula no this is

05:08

actually given to you on your formula

05:09

sheet here at the time of its maximum

05:13

positive velocity positive velocity

05:19

would be right here okay because again

05:23

this is when it's at equilibrium so this

05:27

is when it's at equilibrium but this is

05:30

when it's fully stretched so it's right

05:33

here then you comes back here to

05:35

equilibrium it's it's fully stretch here

05:38

it's fully stretch right here fully come

05:46

press and this is at equilibrium right

05:53

here this is also at equilibrium okay so

06:03

here the time of its maximum positive

06:05

velocity would here would be at three

06:07

seconds now the time of its maximum

06:11

negative velocity is where right here it

06:15

would be at negative velocity because

06:17

look look at it this is its slope here

06:21

is negative the slope here positive

06:26

right positive velocity negative

06:28

velocity wouldn't be occur here at one

06:31

second is this the only spot okay it's

06:35

also at right here do you see how this

06:37

repeats

06:38

right here this lease it's going down so

06:43

the slope here is also negative so you

06:46

could also say five at five second as

06:48

well that time when velocity equals to

06:51

zero okay

06:52

so velocity equals to zero so this is

06:57

when velocity equals to zero so think

06:59

about this as flat slope okay

07:07

so let's let me make that green okay we

07:10

know that occurs where here at zero

07:15

seconds okay then we have it here two

07:21

seconds then we have here at four

07:25

seconds and we have here at six seconds

07:31

these are all when velocity equals to

07:34

zero time of the maximum positive

07:36

acceleration you want to look at it's

07:39

positive acceleration okay so look about

07:43

where it's touching this um it's going

07:46

inwards

07:47

okay so here okay positive so this is

07:52

going in what direction here it's here

07:58

positive alright this is occurring at

08:02

two seconds and here it's also occurring

08:05

at six seconds it's positive it's going

08:08

inwards okay it's point towards the

08:11

inside of the circle remember Center Pro

08:13

celebration at two seconds and six

08:17

seconds negative acceleration so it's

08:20

going down here the acceleration here is

08:23

going to be negative and the

08:25

acceleration here is also negative so

08:28

here that occurred at zero seconds and

08:31

that was also at four seconds

08:33

tie when acceleration is equal to zero

08:38

alright so look at when it equals to

08:40

zero so here alright it's not going

08:44

anywhere here it's not going anywhere

08:46

and here it's not going anywhere so that

08:49

point

08:51

one second that is also when it was

08:54

three second and that is also at five

08:57

seconds okay so there you go that's how

09:00

it would look like good okay now some

09:06

student uses this equation and if you

09:09

seen the lecture notes you should see

09:11

how they get this this is the position

09:14

is defined by the cosine wave and we

09:18

want to write this alright so what is

09:20

our amplitude okay

09:23

our aptitude here is going to define by

09:28

what let's take a look okay how far did

09:35

it go up it goes up to 2 all the way to

09:37

4

09:47

two pi F with 0.25 or you can write that

09:53

as 0.25 or you can write that as 1/4

09:58

however you want to write it and we have

10:01

T okay so we could write down the

10:06

formula x equals to 4 cosine parentheses

10:12

2 pi you can do 0.25 over 1/4 so you

10:16

could do 1/4 here then we have T you can

10:21

rewrite this x equals 2 for cosine

10:25

what's 1/4 of - I think this becomes 1/2

10:30

so this is PI over pi over 1/2 T okay

10:38

you just multiply 2 times 1/4 if that's

10:40

how we got that all right we got this

10:43

information from the chart a straight

10:47

another student graphs and but they have

10:51

this another string collects the

10:53

following data from the same exact thing

10:55

but they got this graph what is the

10:57

difference look at what's the difference

10:59

from this one and this one notice their

11:04

starting value the starting value here

11:08

is what 0 okay what does this mean this

11:13

is at equilibrium so you can just say

11:19

here okay that the students here they

11:25

started when it was fully compressed but

11:27

these students started at equilibrium

11:29

that's what you can write all right so

11:35

what I wrote is the students from Part A

11:37

start their motion sensor when the cart

11:38

was fully compressed their starting

11:40

value on the graph had a value of 4 you

11:43

can see you there

11:44

value of for the students from Part C

11:49

started their motion sensor when the

11:52

cart was at equilibrium equilibrium

11:53

that's why their son position had a

11:55

value of 0 but it behaves the same exact

11:57

way it's the same cart but it starts

12:01

differently okay the fact that this was

12:05

starting from zero this will actually

12:07

change the equation okay

12:10

it has everything is the same so let me

12:13

grab this from this equation but it's

12:22

actually no longer a sign it's no longer

12:24

a cosine wave if this is considered a

12:26

sine wave okay so it's gonna be the same

12:30

way but now it's gonna be sine all right

12:33

X is equal to four same thing but now

12:37

it's sine PI squared over T I PI over

12:43

two times T so it's the same thing it's

12:46

just that this is called a sine wave

12:49

right this is considered a sine wave all

12:52

right so now the second group repeats

12:56

the same procedure thinking that perhaps

12:58

if they add mass to the car it would

13:00

help their analysis on the graph in Part

13:03

C above sketch what position and time

13:05

graph would look like for the car with

13:07

four of its mass okay so how does mass

13:11

affect this so I would like for you to

13:14

actually look at these notes right here

13:16

from what I wrote write the period T

13:20

okay

13:21

the period T is given by T equals to two

13:24

pi square root M over K you would see

13:27

that the larger M the longer the period

13:30

be okay and stiffer the spring K the

13:33

shorter the period good this makes sense

13:36

since

13:44

okay this makes sense since a larger

13:47

mass means more nature therefore slower

13:49

response so how would this one look like

13:52

okay if M increases what happens to the

13:59

period okay all right

14:05

should look like this okay so let's make

14:11

this purple okay so from this time how

14:15

does it look like

14:16

so it's going to start at the same spot

14:19

okay but now it's going to take longer

14:23

for it to go to its trough to its peak

14:27

because again it has more inertia to

14:30

move okay then it's the least decay boom

14:39

all right again why it has a longer the

14:51

period larger the mass okay so think

14:55

about it numerically what happened okay

14:57

T equals to 2 pi square root M over K

15:02

this F goes if if goes up then this

15:12

period goes up good now

15:24

okay so if mass increases peared

15:29

increases what about care what about

15:37

care if K increases what happens to the

15:43

T numerator so this goes down okay all

15:51

right

15:51

there you go those are all your

15:53

solutions for 6c