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The document covers the topic of oscillations, including definitions and examples of periodic and oscillatory motion, as well as the relationship between periodic time and frequency. It explains simple harmonic motion (SHM) as the projection of uniform circular motion and discusses the equations for kinetic energy, potential energy, and total energy in SHM. Additionally, it includes graphical representations of energy versus displacement and time, along with derivations of key formulas related to SHM.
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0% found this document useful (0 votes)
70 views13 pagesOscillations - GSEB
The document covers the topic of oscillations, including definitions and examples of periodic and oscillatory motion, as well as the relationship between periodic time and frequency. It explains simple harmonic motion (SHM) as the projection of uniform circular motion and discusses the equations for kinetic energy, potential energy, and total energy in SHM. Additionally, it includes graphical representations of energy versus displacement and time, along with derivations of key formulas related to SHM.
Uploaded by
rushirajzala2008Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF or read online on Scribd
/ 13
AAKASH SG HIGHWAY
CH 13
Oscillations
‘Section A
eo 1M
the answer of the following questions, [Each carries 3 Marks] [30]
1. What is periodic motion ? Give its examples.
What is oscillatory motion ? Give its examples.
3. Write the defination, unit, dimensional formula and relation between periodic time and frequency.
4. Explaining reference particle and reference circle show that the simple harmonic motion is the
projection of uniform circular motion on a diameter of the reference circle.
5. Obtain the velocity of projected particle executing uniform circular motion.
Obtain instantaneous velocity of a particle executing SHM.
as
Obtain the formula of acceleration of a particle from the formula of displacement of SHM
*
Write the force law for SHM and obtain formula of period of SHM particle.
9. Explain and draw the graphs of kinetic energy, potential energy and mechanical energy versus
displacement for SHM.
10. Explain and draw the graphs of kinetic energy, potential energy and total energy as a function of
time.
Section B
e Ww
the answer of the following questions. [Each carries 5 Marks] (10)
41. Obtain the expressions of kinetic energy, potential energy and total energy in simple harmonic
motion. Also write total energy depends and does not depends on what ?
12, What is simple pendulum ? Deduce an expression for the time period of simple pendulum.
© scanned wth oKEnscamer AAKASH SG HIGHWAY
CH 13
Oscillations
[Section A,
© Write the answer of the following questions, (Each carries 3 Marks} [30]
1. What is periodic motion ? Give its examples.
“+ Ifa body repeats its motion along a certain path, about a fixed point, at a definite interval of time
it's motion is said to have periodic motion.
For examples :
(1) An insect climbs up a ramp and falls down it comes back to the initial point and repeats the
process identically.
“= If we draw a graph of its height above the ground versus time, it would look like below figure
(a)
x10)
es ss,
@
(2) Ifa child climbs up a step, comes down and repeats the process, its height above the ground would
look like that in below figure (b).
x0)
jJf\ f\
ad t
wo
(3) When one play the game of bouncing a ball of the ground, between your palm and the ground,
its height versus time graph would look like as in below figure (c).
x0
r
©
‘= Both the curved parts in figure (c) are sections of a parabola given by the Newton's equation of
motion.
h
wah gt? for downward motion, and
h
dg? for upward motion.
"= Here in both different value of U in each case. Moreover following are also the examples of
periodic motion
"= Motion of needle of sewing machine.
"= Motion of piston of
ilway engine.
"= Revolution of moon around the earth (motion along uniform circular path), "+ Motion of the wings of flying birds ete.
Hence, simple defination of periodic motion is as below :
“A motion that repeats itself at regular intervals of time is called periodic motion.”
"= For such type of motion no restoring force is produced on body (system) hence for continuing such
motion external force is constantly needed.
“All the periodic motion need not to be oscillatory motion.
For example : Circular motion is a periodic motion but it is not oscillatory.
2 Whats oscillatory motion ? Give its examples.
"Ifa body moves to and from back and forth or up and down about a fixed point in a definite interval
of time, such a motion is called an oscillatory motion.
"= — Whenan object in periodic motion is in equilibrium at some point in ts position. When asmall external
force applied on it in this situation a restoring force is occurs in it so that force brings object from its
rium position. Asa result oscillation arise.
"= For example :
() A ball placed in a bow! will be in equilibrium at the bottom. If displaced a litle from the point,
it will perform oscillations in the bowl.
(2) The motion of the pendulum of a wall clock is oscillatory motion.
(3) The motion of a loaded spring, when the load attached to the spring is palled once a little from
its mean position and left to itself.
"Every oscillatory motion is periodic, but every periodic motion need not be oscillatory.
"= Restoring force is produced in such type of motion hence no external force constantly needed for
continuation this motion.
= Such type of motion is also known as harmonic motion.
3. Write the defination, unit, dimensional formula and relation between periodic time and frequency.
= Any motion that repeats itself at regular intervals of time is called periodic motion.
"= “The smallest interval of time after which the motion is repeated is called its period.”
OR
“The time required to complete one oscillation is called periodic time.”
"It is denoted by “I”. Its SI unit is second (s)..
"+ If periodic motion is very fast then the periodic time of its oscillation is expressed in (us). As for
example, periodic time of vibration of quartz crystal.
"= If periodic motion is very slow means time period of revolution of mercury is 88 earth days,
“Periodic time of Halley's comet is 76 years.
"Frequency : The number of oscillation completed in one second is called frequency. It is denoted
by "v" or “/” sign.
"Its SI unit is “s!” or Hz (Hertz). In the honour of the Heich Rudolph Hertz (1857-1894)
1 Hz = 1 oscillation per second
“= Frequency v need not to be an integer.
“Dimensional formula of frequency is |M°L°T-!],
Angular frequency : “2n tie the frequency of an oscillator is called angular frequency’.
‘OF scanned ws OREN Scanner © = 2nf and its unit is rads“!
dimensional formula = [M° L°T-!]
4. Explaining reference particle and reference circle show that the simple harmonic motion is the
projection of uniform circular motion on a diameter of the reference circle.
"+ Suppose as shown in figure, a particle is moving with a constant angular speed @ in an anti-
clockwise direction on a circular path having centre O and radius A.
= Attime ¢= 0, particle at position P, and its position vector OP; makes an angle @ with positive X-
axis.
= Projection of OP, on X-axis is OP".
"> Attime = 1, particle displaced by angular displacement wt,
reaches at point P, and its position
vector OP, makes an angle wf + with X-axis.
Projection of position vector OP, on X-axis is OP’,
"= As the particle P moving on a circle, its point of perpendicular on X-axis is given by x(t) = Acos(wt
+4). It is the X-component of position vector at any time,
“This equation is the general equation of SHM.
From this it is said that the projection of uniform circular motion on a diameter of the reference
circle is SHM.
"The particle moving on uniform circular path is called reference particle and the circular path of
reference particle is reference circle.
"If the projection of reference particle taken on Y-axis, the displacement of particle on Y-axis is y(0)
= Asin(at + §).
Obtain the velocity of projected particle executing uniform circular motion.
“+ The linear velocity of a particle moving in a circular path of radius A with angular speed w is the
radius time to angular velocity.
2 v=ho
"+ The direction of linear velocity of a particle at time f is in the direction of tangent at a point on
the circle at that time.
© scanned wt onEN Scanner hbo
?
le
According to figure, two mutual components of linear velocity Aw, component Acsin(ct + 4) in X-
axis direction, is the velocity of projection on X-axis of a particle at time t.
~Aasin(or + @)
Direction of v(t)is opposite to X-axis and itis the instantaneous velocity of particle executing SHM.
Velocity of particle P' is v(4) in the reference of projection of pat
2 wt)
Obtain instantaneous velocity of a particle executing SHM.
Instantaneous velocity of SHM particle is the rate of change of displacement for a time interval.
‘Suppose the displacement of SHM particle at time t with amplitude A and angular speed o is
x(0) = Acos(wt + 4) = @
where @ is the initial phase
Differentiating equation (1) with respect to f gives the instantaneous velocity
x(t)
ut) = Se
a
qiAeostet + 81
~Ausin(or + 4)
yi cot oreo
- Aay(t/1=c0s"(or+ 6}
oof? =A? c0s"(or +6)
sofa?)
Generally » = tw/A?—x?
Special Cases :
but sin(ot +9)
&
1"
) At mean position, x = 0
velocity v = 0A
In +x-direction, v is positive and in -x-direction, v is negative.
Hence, maximum velocity of SHM particle tyyye = Aw
(2) At extreme point, for velocity x =| Al,
v = toa? -A® [lar = al
v=0
© scm in oe Saner «+ The velocity of SHM particle at extreme points is zero.
Obtain the formula of acceleration of a particle from the formula of displacement of SHM.
‘Acceleration of SHM particle is two times differentiation of displacement wr. to time of particle.
OR
Acceleration of SHM particle is the differentiation of velocity of particle w.rt. time
The displacement of SHM particle at time 1,
x(t) = Acos(ot + 6)
Differentiating with time 1,
(9) = -Aasin(or + 6)
Again differentiating with
a(t) = -AuPcostor + 9)
alt) = - 0? x0) oD
~Acos(at + ¢) = x0]
generally @ = - a?
Special cases :
() At mean position x(2)
ao x0
a
na=0
Acceleration at mean posi
in is zero and velocity is maximum,
2) At extreme points, x(t) = |A|
a= -o?|A]
2 @ = @A or a= WA
Hence, at extreme points acceleration is maximum is called dma
Ls yy = Aa?
Acceleration in +x-direction is negative and in-x-direction it is positive,
Write the force law for SHM and obtain formula of period of SHM particle.
‘The acceleration of SHM particle
alt) = - oxi.)
where x(t) is the displacement at time t
‘The force exerted on SHM particle according to Newton's second law
F = matt)
oF = -morxto) = Q)
[From equation (1)]
‘The force exerted on SHM is always directly proportional to displacement and towards the mean
position. This force is restoring force for SHM
2 F «=x
‘© scanned ws OREN Scanner 2 F = -kx(t) @)
where k is force constant of SHM and is the restoring force per unit displacement comparing
equation (2) and (3)
k= mot
E
so= ft
m
Putting @ = T’
ri
rem
9. Explain and draw the graphs of kinetic energy, potential energy and mechanical energy versus
displacement for SHM.
"+ Kinetic energy of SHM K(x) = RAP 3)
2
Potential energy U(x) = fee and
Mechanical energy E = Frat
= The magnitude of energies at different position of SHM are shown as below :
sr. Kinetic energy | Potential energy |Mechanical energy E = K + U|
JX Displacement] x(x) = Fen —x%)| cay = Pe?
@] x=4A 0 peat
@) x=+x SA? =x?) phe i
@ 0 pat
@} x=-x dix? ka?
© D Tae Tia?
"From these values graph of energies versus displacement is obtained as below :
KG) + Uta)
= Following points are clear from graph :
(© scanned wt OREN Scanner 10.
(1) Ex graph is linear and is parallel to displacement axis, so we can say that mechanical energy
does not depends on displacement, but remains constant.
(2) Shapes of K(x) +x and UG) > x graphs are parabolic.
(3) At fixed point x = 0, potential energy is zero and kinetic energy is maximum and is equal to
mechanical energy.
(4) As the oscillator goes towards any side increase in potential energy is equal to decrease in
kinetic energy.
(6) At extreme point (y = |A)), potential energy is maximum and kinetic energy is zero. This
‘maximum potential is equal to mechanical energy.
(6) At any point of SHM path, the sum of potential and kinetic energy is equal to mechanical
energy.
( Coordinates of point of intersections of graph of kinetic energy and potential energy is
AE
te, 5].
(5)
Explain and draw the graphs of kinetic energy, potential energy and total energy as a function of
time.
Kinetic energy of SHM particle
1
Kit) = —kA*sin*(or +6)
2
Potential energy U(1) = 2 cos*(wt +o) and
2
Mechanical energy E = Kio + U()
Lat sin’ 2,
+ Fka*[sin*tor + 9)+ costar + 9)]
Lag?
z
Here, in table values of kinetic, potential and total energy in terms of period of SHM particle is
shown and by using these values a graph of a energy versus time is shown,
Sr. Kinetic energy Potential energy | Mechanical energy
No] Time | x(e) = Fka®sin*or +4)] Ule) = kA cos*ar+9)| B= KO + UO
ao peat 0 peat
a} 4 o Py pea?
@ = 0 peat
| = 0 bea? heat
@| T 0 fiat ML.
E:
K+,
KO
o T T
2
Following points are clear from this graph :
(Q) In SHM both kinetic and potential energy are positive hence total energy is also positive.
(2) In SHM, kinetic energy becomes two time maximum in every period and becomes two times
zero. This maximum kinetic energy is equal to the total energy.
(3) In SHM, potential energy becomes two times maximum in every period and become two time
zero. This maximum potential energy is equal to the total energy.
T
(4) The increase or decrease in kinetic energy from energy time of = in SHM is equal to decrease
or increase in potential respectively.
(5) At any time the sum of kinetic and potential energy is constant and it is mechanical energy.
Hence mechanical energy does not depend on time.
[Section B
Write the answer of the following questions. [Each carries 5 Marks) (10)
Obtain the expressions of kinetic energy, potential energy and total energy in simple harmonic
motion. Also write total energy depends and does not depends on what ?
Kinetic energy : Displacement of SHM particle at any instant
X= Acos(ot + 4)
where A= amplitude, » = angular frequency
velocity v a2 Atacostor + on
2 v = —Aosin(ot + 4)
Now kinetic energy,
1
K = 1m?
7™
= pinta? sin*or +9)
2k FAtsin2 (or +9) ve () [2 me? = ky
Hence, it is a periodic function of time, being zero when the displacement is maximum and
maximum when the particle is at the mean position.
T
The period of kinetic energy is >.
Potential energy : Potential energy is possible only for conservative forces and restoring force is the
conservative force in SHM.
If the restoring force exerted on a particle of SHM from its mean position is x then
F=-ke
(© scanned wt OREN Scanner "+ The work opposite to restoring force for small displacement dx of a particle is,
dW = -Fdx
dW = —kxdx
"+ Now work done for displacement of particle from x = 0 tox
we jaw
The work done opposite to restoring force is stored as potential energy in particle.
+. potential energy in displacement x of particle is
U=-w
A gt
JraFatcostar +9)
4 U = FRAP costar +4)
Hence, the potential energy of a particle executing simple harmonic motion is also periodic.
At mean position potential energy is zero and at extreme points it is maximum, Hence, the period
Tr
of potential energy is 5.
Total energy : The sum of kinetic and potential energy of SHM is known as total energy E.
2 E=K+U
b= Hewtenttrs ye deateosat +6)
Apa’[sin’ 2
B= Shs [sin’ (at + 9) + cos’ (ot+9)]
sin*(wt +9)+ cost +9) = 1]
dL inw2A2
or E =