Class 11 Exam > Class 11 Videos > Simple Harmonic motion ": Displacement time graph - Oscillations
Simple Harmonic motion ": Displacement time graph - Oscillations Video Lecture - Class 11
FAQs on Simple Harmonic motion ": Displacement time graph - Oscillations Video Lecture - Class 11
| 1. What is simple harmonic motion? |  |
Ans. Simple harmonic motion refers to the repetitive back-and-forth motion of an object around a stable equilibrium position. It occurs when the restoring force acting on the object is directly proportional to its displacement from the equilibrium position and acts in the opposite direction.
| 2. How can the displacement-time graph be used to analyze oscillations? | |
Ans. The displacement-time graph provides a visual representation of the motion of an object undergoing simple harmonic motion. It shows how the displacement of the object changes over time. By analyzing the shape of the graph, we can determine the amplitude, period, frequency, and phase of the oscillation.
| 3. What does the shape of the displacement-time graph reveal about the oscillation? | |
Ans. The shape of the displacement-time graph reveals important characteristics of the oscillation. The amplitude is determined by the maximum displacement from the equilibrium position, while the period is the time taken to complete one full oscillation. The frequency is the number of oscillations per unit time, and the phase indicates the position of the object in its oscillatory cycle at a given time.
| 4. How can the displacement-time graph be used to calculate the frequency of oscillation? | |
Ans. The frequency of oscillation can be calculated by determining the time taken for one complete oscillation, which corresponds to the period. The frequency (f) is then the reciprocal of the period (T), given by the formula f = 1/T. By measuring the time interval between consecutive peaks or troughs on the displacement-time graph, we can compute the frequency.
| 5. What is the significance of the equilibrium position in simple harmonic motion? | |
Ans. The equilibrium position in simple harmonic motion is crucial as it represents the position where the net force acting on the object is zero. When displaced from the equilibrium position, a restoring force is exerted on the object, pulling it back towards the equilibrium position. The motion of the object is characterized by its oscillation around this stable equilibrium point.
Text Transcript from Video
hello friends this video on oscillations
part 6 is brought to you by exam feel
calm no hope here from exam please make
sure that you have watched all the
videos from part 1 to part 5 before
going ahead with part 6 before going
into the new terms which we heard now
let us try to discuss the displacement
in case of simple harmonic motion I have
already spoken about displacement in the
last few slides now let us see how does
displacement value with time I mean
mathematically I have shown you how does
it value with time it is basically it
basically values as a cosine function
now let us see the same graphically
let us see how displacement changes with
time with a help of a graph so in order
to plot a graph we should be able to
know the values of displacement at
different intervals of time so now let
before we go and plot the graph let us
calculate the values of displacement at
various instants of time so how this
displacement represent it it is
represented as X of T is equal to a cos
Omega T plus Phi right as of now let us
consider that the phase constant that is
Phi is equal to 0 why am i considering a
zero that that will become clear to you
when I explain what the space Const so
here I am considering a cos Omega T so
this is my displacement
now with the help of this let us
calculate the values of displacement at
various instance first let us calculate
its value at the mean position let us
think from where does the motion starts
let us suppose this is your pendulum so
if you do not apply any force on the
controller it will not move on its own
right so it will be in its name musician
now let us suppose what do you do you
take the pendulum you take this mask
take it to this extreme end and then
leave it you take it in here
and then leave it what will happen it'll
start swinging it will start moving this
way and the motion will keep continuing
and then gradually it'll slow down and
come down to its main position like so
from there did your motion stop the
motion started from this extreme end so
for this motion to start you had to drag
this mass to the extreme end otherwise
it would have not started on its own
right so at T is equal to zero that
means at the very initial time where was
this mask this was at an extreme
position so let us calculate the
displacement at T is equal to C now
according to this expression at T is
equal to zero X of 0 will be equal to a
cos 0 so that will be equal to a because
cos 0 is 1 so this is the value of
displacement at time T is equal to C
right now let us divide we know that the
total time period is T correct so let us
divide the prototype period into smaller
intervals let us suppose the motion is
starting from here
so here time T is equal to 0 correct
so here T is equal to 0 now this
friend.you this mass comes back from
here it processes the mean position so
by this time what is the time at this
point it is nothing but D by 4 at this
point then again the mass goes to this
extreme end now what is the time at this
point it is nothing but T by 2 because
it takes another T by 4 to reach here
after that it again comes back to the
mean position that time what is the time
that is nothing but D by 2 plus T by
pouring it which is nothing but 3 T by 4
and then again it goes back to this
in position then the time is nothing but
T right so this is one time period as I
told you after this again the same
pattern will start repeating so let us
calculate the values of the displacement
at each of these time intervals that is
at T is equal to 0 at T is equal to T by
4 T by 2 3 T by T 3 T by 4 and capital T
so now next let us calculate when the
time T is equal to capital T by 4 that
means one-fourth of the time period like
understand this once again so that you
don't get confused one time period means
let us suppose this is main position
this is an extreme position this is
another extreme position now we know
that 1 capital T that is one time period
is the time taken to complete one cycle
so your cycle means if your motion
starts from this position so this
pendulum will come till this extreme int
and then it will go back to this extreme
end so the total time taken for this is
1 capital T so what I am doing is I am
dividing it into four intervals right
first it is coming from here to mean
position again from being position to
extreme and extreme and back to mean
position and from being position back to
the starting point so we can see that
this is if this is T is equal to 0 then
what is this this is T is equal to T by
4 then what does this this is T is equal
to T by 2 what does this this t is equal
to 3 T by 4 and this is T is equal to
capital T here so now you're clear what
all time periods advertising videos we
are calculating the value of
displacement now let us come back to the
calculation of displacement so XT is
equal to T by 4 where I will be when T
is equal to T by 4
we are at the main position right so
here the value of displacement will be X
T by 4 is equal to a cos Omega T by 4
now what does Omega as per definition of
time period we have already seen that
Omega is nothing but our capital T is
nothing but 2 pi by Omega so using this
we can see that Omega is equal to 2 pi
by T right now if you see there this get
cancelled so a cost I by 2 cos Phi by 2
is 0 so this value comes out to be C so
that means that this placement at time T
is equal to T by 4 is equal to 0 now let
us calculate the same at t is equal to t
by 2 at t is equal to t by 2 it is again
at the extreme position so there X of T
by 2 is equal to a cos Omega T by 2 so
here we can write a cos 2 pi by T into T
by 2 so this TT will get answer to 2
will get cancelled so this comes out to
be e cos Phi cos Phi is equal to minus 1
so this comes out to be minus a right
now let us go so till now we have
calculated T is equal to 0 T is equal to
T by 4 T is equal to D by 2 now let us
calculate for T is equal to 3 T by 4
now T is equal to three T by 4 so here
now the object is at main position again
so at this mean position the value of x
at 3 T by 4 will be equal to a cos Omega
C T by 4
now we put the value of Omega as 2 pi by
T into 3 T by 4 so this gets cancelled
so we get it as a cos 3 Phi by 2 now it
is 3 PI by 2 can be written as 4 so 5
plus Phi by 2
now using the formula of course a plus B
you see that this value comes out to be
0 so this is a name comes out to be C
now again let's look at the next
position that is at T is equal to
capital T there the object is again at
the extreme end from where it started so
it's a capital T will be equal to a cos
Omega T so Omega is nothing but 2 pi T
into T sub T gets cancelled so we get a
cos 2 pi so cos 2 pi easily and 1 so we
get this value as e again now with the
help of these values which we calculated
just now that is that is just let us
just actually the values which we
calculated the values of T versus the
values of X so at time T is equal to 0 T
by 4 T by 2 3 T by 4 and T for these
values what are the values of
displacements that we got displacement
for at T is equal to zero was eight at T
by four was 0 for T by 2
it was minus 8 sweetie mile for it was
again 0 and 14 it was again a so this is
how the displacement valid so now with
the help of this data we can very easily
plot them laugh right so let us plot the
graph so this is your just placement let
us suppose we consider time around its
axis is time T around x axis and your
displacement that is X or T along the y
axis right now let us suppose let us
consider at this point to is that what a
let us say this is a this point
represents a if that is a in this point
must represent minus a so it's travel as
T is equal to 0 that this time is equal
to 0 displacement is a so this is my
first point to plot the graph at T is
equal to T by 4 now let us suppose if
this is capital T let us plot this way
let us suppose this is T by 4 this is T
by 2 this is TT by 4 and this is capital
T so I will be easy for us to plot it
so at T by 4 the value is 0 so this is
our next point T by 2 it is minus a at T
by 2 it is
a change here at 3 T by 4 it is again 0
so this point at T it is again a so that
means at 20 it is again at value a so
these are the five points which we found
now let us try to join his points
you
and then this third keeps repeating
because we calculated the values for one
cycle rent we calculated the values of
displacement from t is equal to 0 to t
is equal to capital t now after capital
t the same motion keeps repeating right
so that means this curve will also keep
repeating so again you get a curve like
this so this is how your displacement
versus time curve will look like right
so we see that the displacement versus
time curve for a simple harmonic motion
is a sinusoidal curve that is the sine
or a cosine curve which varies
periodically with time
thank you please visit exam fee or comm
to watch free educational videos try 3
online tests get the best quality study
materials study from the best tutors and
mentors and much more thank you once
again
part 6 is brought to you by exam feel
calm no hope here from exam please make
sure that you have watched all the
videos from part 1 to part 5 before
going ahead with part 6 before going
into the new terms which we heard now
let us try to discuss the displacement
in case of simple harmonic motion I have
already spoken about displacement in the
last few slides now let us see how does
displacement value with time I mean
mathematically I have shown you how does
it value with time it is basically it
basically values as a cosine function
now let us see the same graphically
let us see how displacement changes with
time with a help of a graph so in order
to plot a graph we should be able to
know the values of displacement at
different intervals of time so now let
before we go and plot the graph let us
calculate the values of displacement at
various instants of time so how this
displacement represent it it is
represented as X of T is equal to a cos
Omega T plus Phi right as of now let us
consider that the phase constant that is
Phi is equal to 0 why am i considering a
zero that that will become clear to you
when I explain what the space Const so
here I am considering a cos Omega T so
this is my displacement
now with the help of this let us
calculate the values of displacement at
various instance first let us calculate
its value at the mean position let us
think from where does the motion starts
let us suppose this is your pendulum so
if you do not apply any force on the
controller it will not move on its own
right so it will be in its name musician
now let us suppose what do you do you
take the pendulum you take this mask
take it to this extreme end and then
leave it you take it in here
and then leave it what will happen it'll
start swinging it will start moving this
way and the motion will keep continuing
and then gradually it'll slow down and
come down to its main position like so
from there did your motion stop the
motion started from this extreme end so
for this motion to start you had to drag
this mass to the extreme end otherwise
it would have not started on its own
right so at T is equal to zero that
means at the very initial time where was
this mask this was at an extreme
position so let us calculate the
displacement at T is equal to C now
according to this expression at T is
equal to zero X of 0 will be equal to a
cos 0 so that will be equal to a because
cos 0 is 1 so this is the value of
displacement at time T is equal to C
right now let us divide we know that the
total time period is T correct so let us
divide the prototype period into smaller
intervals let us suppose the motion is
starting from here
so here time T is equal to 0 correct
so here T is equal to 0 now this
friend.you this mass comes back from
here it processes the mean position so
by this time what is the time at this
point it is nothing but D by 4 at this
point then again the mass goes to this
extreme end now what is the time at this
point it is nothing but T by 2 because
it takes another T by 4 to reach here
after that it again comes back to the
mean position that time what is the time
that is nothing but D by 2 plus T by
pouring it which is nothing but 3 T by 4
and then again it goes back to this
in position then the time is nothing but
T right so this is one time period as I
told you after this again the same
pattern will start repeating so let us
calculate the values of the displacement
at each of these time intervals that is
at T is equal to 0 at T is equal to T by
4 T by 2 3 T by T 3 T by 4 and capital T
so now next let us calculate when the
time T is equal to capital T by 4 that
means one-fourth of the time period like
understand this once again so that you
don't get confused one time period means
let us suppose this is main position
this is an extreme position this is
another extreme position now we know
that 1 capital T that is one time period
is the time taken to complete one cycle
so your cycle means if your motion
starts from this position so this
pendulum will come till this extreme int
and then it will go back to this extreme
end so the total time taken for this is
1 capital T so what I am doing is I am
dividing it into four intervals right
first it is coming from here to mean
position again from being position to
extreme and extreme and back to mean
position and from being position back to
the starting point so we can see that
this is if this is T is equal to 0 then
what is this this is T is equal to T by
4 then what does this this is T is equal
to T by 2 what does this this t is equal
to 3 T by 4 and this is T is equal to
capital T here so now you're clear what
all time periods advertising videos we
are calculating the value of
displacement now let us come back to the
calculation of displacement so XT is
equal to T by 4 where I will be when T
is equal to T by 4
we are at the main position right so
here the value of displacement will be X
T by 4 is equal to a cos Omega T by 4
now what does Omega as per definition of
time period we have already seen that
Omega is nothing but our capital T is
nothing but 2 pi by Omega so using this
we can see that Omega is equal to 2 pi
by T right now if you see there this get
cancelled so a cost I by 2 cos Phi by 2
is 0 so this value comes out to be C so
that means that this placement at time T
is equal to T by 4 is equal to 0 now let
us calculate the same at t is equal to t
by 2 at t is equal to t by 2 it is again
at the extreme position so there X of T
by 2 is equal to a cos Omega T by 2 so
here we can write a cos 2 pi by T into T
by 2 so this TT will get answer to 2
will get cancelled so this comes out to
be e cos Phi cos Phi is equal to minus 1
so this comes out to be minus a right
now let us go so till now we have
calculated T is equal to 0 T is equal to
T by 4 T is equal to D by 2 now let us
calculate for T is equal to 3 T by 4
now T is equal to three T by 4 so here
now the object is at main position again
so at this mean position the value of x
at 3 T by 4 will be equal to a cos Omega
C T by 4
now we put the value of Omega as 2 pi by
T into 3 T by 4 so this gets cancelled
so we get it as a cos 3 Phi by 2 now it
is 3 PI by 2 can be written as 4 so 5
plus Phi by 2
now using the formula of course a plus B
you see that this value comes out to be
0 so this is a name comes out to be C
now again let's look at the next
position that is at T is equal to
capital T there the object is again at
the extreme end from where it started so
it's a capital T will be equal to a cos
Omega T so Omega is nothing but 2 pi T
into T sub T gets cancelled so we get a
cos 2 pi so cos 2 pi easily and 1 so we
get this value as e again now with the
help of these values which we calculated
just now that is that is just let us
just actually the values which we
calculated the values of T versus the
values of X so at time T is equal to 0 T
by 4 T by 2 3 T by 4 and T for these
values what are the values of
displacements that we got displacement
for at T is equal to zero was eight at T
by four was 0 for T by 2
it was minus 8 sweetie mile for it was
again 0 and 14 it was again a so this is
how the displacement valid so now with
the help of this data we can very easily
plot them laugh right so let us plot the
graph so this is your just placement let
us suppose we consider time around its
axis is time T around x axis and your
displacement that is X or T along the y
axis right now let us suppose let us
consider at this point to is that what a
let us say this is a this point
represents a if that is a in this point
must represent minus a so it's travel as
T is equal to 0 that this time is equal
to 0 displacement is a so this is my
first point to plot the graph at T is
equal to T by 4 now let us suppose if
this is capital T let us plot this way
let us suppose this is T by 4 this is T
by 2 this is TT by 4 and this is capital
T so I will be easy for us to plot it
so at T by 4 the value is 0 so this is
our next point T by 2 it is minus a at T
by 2 it is
a change here at 3 T by 4 it is again 0
so this point at T it is again a so that
means at 20 it is again at value a so
these are the five points which we found
now let us try to join his points
you
and then this third keeps repeating
because we calculated the values for one
cycle rent we calculated the values of
displacement from t is equal to 0 to t
is equal to capital t now after capital
t the same motion keeps repeating right
so that means this curve will also keep
repeating so again you get a curve like
this so this is how your displacement
versus time curve will look like right
so we see that the displacement versus
time curve for a simple harmonic motion
is a sinusoidal curve that is the sine
or a cosine curve which varies
periodically with time
thank you please visit exam fee or comm
to watch free educational videos try 3
online tests get the best quality study
materials study from the best tutors and
mentors and much more thank you once
again
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Nov 08, 2024 Last updated |
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