AP Physics Workbook 6.C Equations of Motion for Simple Harmonic Motion
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(Οt) and x = A sin(Οt), to describe the system's behavior, highlighting how different starting conditions can lead to different representations of the motion.
Takeaways
- π The topic is AP Physics, focusing on simple harmonic motion and equations of motion.
- ποΈ A cart of mass M on a smooth surface is attached to an ideal spring and undergoes simple harmonic motion.
- π The motion detector collects data to create a graph of position vs. time for analysis.
- π Frequency is the number of cycles per second, measured in Hertz (Hz), and wavelength is the distance from peak to peak.
- π 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.
- π Amplitude is the maximum displacement from equilibrium, representing the peak and trough of the oscillation.
- π The graph shows that the cart's oscillation completes a full cycle in 4 seconds, resulting in a frequency of 0.25 Hz.
- π€ The angular velocity (Ξ©) is calculated using the formula Ξ© = 2Ο/T, which simplifies to Ξ© = Ο/2 for this scenario.
- π The maximum positive and negative accelerations occur at different times during the oscillation, pointing towards the center of the oscillation path.
- π The position of the cart is described by a cosine wave, with the equation x = A*cos(Οt), where A is the amplitude and Ο is the angular frequency.
- π§ The period T is influenced by the mass (M) and the spring constant (K), following the formula T = 2Οβ(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 = 2Οβ(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 = 2Οβ(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(Οt + Ο)), and a sine wave when the sensor starts at equilibrium (x = A sin(Οt + Ο)), where A is the amplitude, Ο is the angular velocity, t is time, and Ο is the phase constant.
Outlines
π 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.
π 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.
π 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.
π§ 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
Keywords
π‘Simple Harmonic Motion
π‘Frequency
π‘Wavelength
π‘Amplitude
π‘Period
π‘Angular Velocity
π‘Equilibrium Position
π‘Cosine Wave
π‘Sine Wave
π‘Mass
π‘Spring Constant
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(Οt + 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 = 2Οβ(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
all right welcome this is the AP Physics
workbook solution here we have unit six
in part monic motion the section is six
point C equations of motion for simple
harmonic here here's the scenario
a cart of mass M resting on a smooth
surface is attached to an ideal spring
the cart is displacement to the right as
the distance Delta X from equilibrium
and release while the car oscillates
around the equilibrium position a motion
detector collects data and to make the
following graph of position as a
function of time before we read this I
would like to give you some notes all
right so we understand the right vocab
that we're using the frequency is the
number of cycles or the number of
wavelength passing through a position in
one second the units of frequency is
considered car Hertz which means one
second the wavelength is measured from
peak to peak
so from here peak to peak or any okay so
here it would be one complete it would
be consider one wavelength here right so
Springs are like waves and circles okay
here's one period a period T is the time
for one revolution or in case in this
case spring the time for one complete
oscillation or one crest and one trough
one cress one trough and it comes back
here to equilibrium okay oscillations
can be also we call vibrations and
cycles this is the amplitude from
equilibrium into the maximum height is
called the amplitude okay as you go to
the right it's called the equilibrium
it's in the equilibrium time period is
defined as 2 PI over Omega or 1 over f
which is considered the frequency notice
time and frequencies are inverse of each
other okay a more interesting situation
develops when we exert the same force on
the system it repeats in in repetitive
periodic motion this is the simple
harmonic motion
you should see that it is fully
compressed aptitude is its maximum
height here at the amplitude here it's
fully stretch which is at its trough
here do you see how it comes back to its
original position it completes its full
cycle here that's called T the time it
takes to complete one full cycle a
stands for the amplitude and for the
frequency and T is for the period good
you're gonna need these ideas to help
you do this if you would like a more
detailed explanation on the notes please
watch the simple harmonic lecture notes
alright so let's see how long it take
for T for the period so how long did it
take for this to complete one full cycle
so it starts from here and it goes all
the way until it reaches this so it goes
from zero all the way to four so we
could write here four seconds okay
because this is its complete cycle now F
is considered its frequency so you can
just write it right here T is equal to 1
over F and we know that here it is 4 so
you can write you can make the
substitution here T equals to 4 multiply
frequency is equal to 1/4 you can also
write that as 0.25 the units for
frequency is considered like we said
here is call hertz which is per second
so the frequency here is 0.25 Hertz so
point two five of the wave occurred in
one second
that's what this means now W which is
which is its angular velocity okay how
do we get its angular velocity we can
use this formula right here
is equal to 2 PI over Omega now we can
solve for W or Omega okay so T is for
two PI over W cross multiply you get W
Omega is equal to two PI over four which
can be simplified to just PI over two so
you could say here this is PI over two
the units for this is rad over seconds
all right here's the math if you would
like to see it okay do you need to
memorize this formula no this is
actually given to you on your formula
sheet here at the time of its maximum
positive velocity positive velocity
would be right here okay because again
this is when it's at equilibrium so this
is when it's at equilibrium but this is
when it's fully stretched so it's right
here then you comes back here to
equilibrium it's it's fully stretch here
it's fully stretch right here fully come
press and this is at equilibrium right
here this is also at equilibrium okay so
here the time of its maximum positive
velocity would here would be at three
seconds now the time of its maximum
negative velocity is where right here it
would be at negative velocity because
look look at it this is its slope here
is negative the slope here positive
right positive velocity negative
velocity wouldn't be occur here at one
second is this the only spot okay it's
also at right here do you see how this
repeats
right here this lease it's going down so
the slope here is also negative so you
could also say five at five second as
well that time when velocity equals to
zero okay
so velocity equals to zero so this is
when velocity equals to zero so think
about this as flat slope okay
so let's let me make that green okay we
know that occurs where here at zero
seconds okay then we have it here two
seconds then we have here at four
seconds and we have here at six seconds
these are all when velocity equals to
zero time of the maximum positive
acceleration you want to look at it's
positive acceleration okay so look about
where it's touching this um it's going
inwards
okay so here okay positive so this is
going in what direction here it's here
positive alright this is occurring at
two seconds and here it's also occurring
at six seconds it's positive it's going
inwards okay it's point towards the
inside of the circle remember Center Pro
celebration at two seconds and six
seconds negative acceleration so it's
going down here the acceleration here is
going to be negative and the
acceleration here is also negative so
here that occurred at zero seconds and
that was also at four seconds
tie when acceleration is equal to zero
alright so look at when it equals to
zero so here alright it's not going
anywhere here it's not going anywhere
and here it's not going anywhere so that
point
one second that is also when it was
three second and that is also at five
seconds okay so there you go that's how
it would look like good okay now some
student uses this equation and if you
seen the lecture notes you should see
how they get this this is the position
is defined by the cosine wave and we
want to write this alright so what is
our amplitude okay
our aptitude here is going to define by
what let's take a look okay how far did
it go up it goes up to 2 all the way to
4
two pi F with 0.25 or you can write that
as 0.25 or you can write that as 1/4
however you want to write it and we have
T okay so we could write down the
formula x equals to 4 cosine parentheses
2 pi you can do 0.25 over 1/4 so you
could do 1/4 here then we have T you can
rewrite this x equals 2 for cosine
what's 1/4 of - I think this becomes 1/2
so this is PI over pi over 1/2 T okay
you just multiply 2 times 1/4 if that's
how we got that all right we got this
information from the chart a straight
another student graphs and but they have
this another string collects the
following data from the same exact thing
but they got this graph what is the
difference look at what's the difference
from this one and this one notice their
starting value the starting value here
is what 0 okay what does this mean this
is at equilibrium so you can just say
here okay that the students here they
started when it was fully compressed but
these students started at equilibrium
that's what you can write all right so
what I wrote is the students from Part A
start their motion sensor when the cart
was fully compressed their starting
value on the graph had a value of 4 you
can see you there
value of for the students from Part C
started their motion sensor when the
cart was at equilibrium equilibrium
that's why their son position had a
value of 0 but it behaves the same exact
way it's the same cart but it starts
differently okay the fact that this was
starting from zero this will actually
change the equation okay
it has everything is the same so let me
grab this from this equation but it's
actually no longer a sign it's no longer
a cosine wave if this is considered a
sine wave okay so it's gonna be the same
way but now it's gonna be sine all right
X is equal to four same thing but now
it's sine PI squared over T I PI over
two times T so it's the same thing it's
just that this is called a sine wave
right this is considered a sine wave all
right so now the second group repeats
the same procedure thinking that perhaps
if they add mass to the car it would
help their analysis on the graph in Part
C above sketch what position and time
graph would look like for the car with
four of its mass okay so how does mass
affect this so I would like for you to
actually look at these notes right here
from what I wrote write the period T
okay
the period T is given by T equals to two
pi square root M over K you would see
that the larger M the longer the period
be okay and stiffer the spring K the
shorter the period good this makes sense
since
okay this makes sense since a larger
mass means more nature therefore slower
response so how would this one look like
okay if M increases what happens to the
period okay all right
should look like this okay so let's make
this purple okay so from this time how
does it look like
so it's going to start at the same spot
okay but now it's going to take longer
for it to go to its trough to its peak
because again it has more inertia to
move okay then it's the least decay boom
all right again why it has a longer the
period larger the mass okay so think
about it numerically what happened okay
T equals to 2 pi square root M over K
this F goes if if goes up then this
period goes up good now
okay so if mass increases peared
increases what about care what about
care if K increases what happens to the
T numerator so this goes down okay all
right
there you go those are all your
solutions for 6c
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