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Subject: B. Tech. PHYSICS – I (3 – 1 – 0)
Waves and Oscillations
Periodic & Oscillatory Motion:-
The motion in which repeats after a regular interval of time is called periodic motion.
- The periodic motion in which there is existence of a restoring force and the body moves along the same path to and fro about a definite point called equilibrium position/mean position, is called oscillatory motion.
- In all type of oscillatory motion one thing is common i.e each body (performing oscillatory motion) is subjected to a restoring force that increases with increase in displacement from mean position. 3. Types of oscillatory motion:- It is of two types such as linear oscillation and circular oscillation. Example of linear oscillation:- 1. Oscillation of mass spring system. 2. Oscillation of fluid column in a U-tube. 3. Oscillation of floating cylinder. 4. Oscillation of body dropped in a tunnel along earth diameter. 5. Oscillation of strings of musical instruments. Example of circular oscillation:- 1. Oscillation of simple pendulum. 2. Oscillation of solid sphere in a cylinder (If solid sphere rolls without slipping). 3. Oscillation of a circular ring suspended on a nail.
- Oscillation of balance wheel of a clock.
- Rotation of the earth around the sun.
Oscillatory system:-
- The system in which the object exhibit to & fro motion about the equilibrium position with a restoring force is called oscillatory system.
- Oscillatory system is of two types such as mechanical and non- mechanical system. 3. Mechanical oscillatory system:- In this type of system body itself changes its position. For mechanical oscillation two things are specially responsible i.e Inertia & Restoring force. E.g oscillation of mass spring system, oscillation of fluid-column in a U-tube, oscillation of simple pendulum, rotation of earth around the sun, oscillation of body dropped in a tunnel along earth diameter, oscillation of floating cylinder, oscillation of a circular ring suspended on a nail, oscillation of atoms and ions of solids, vibration of swings…etc.
4.Non-mechanical oscillatory system:-
In this type of system, body itself doesn‟t change its position but its physical property varies periodically.
e.g:-The electric current in an oscillatory circuit, the lamp of a body which is heated and cooled periodically, the pressure in a gas through
Where ω^2 =
Here ω=√ is the angular frequency of the oscillation.
Equation (2) is called general differential equation of SHM.
By solving these differential equation
x= + ……… (3)
Where , are two constants which can be determined from the initial condition of a physical system.
Appling de-Moiver‟s theorem
x= cos +isin ) + cos -isin )
x= + ) cos + ) sin
x= Ccos +Dsin ………(4)
Where C = + & D=
Let assume,
C=A
D=Acos
Putting these value in equation (4)
x=A cos +Acos sin
x=A ( cos )
x=A sin ) ……….. (5)
Where A=√ 2 +D^2 ) &
Similarly, the solution of differential equation can be given as
x=Acos ) ………(6)
Here A denotes amplitude of oscillatory system, ) is called phase and is called epoch/initial phase/phase constant/phase angel.
Equation (5) and (6) represents displacement of SHM.
Velocity in SHM:-
=Asin )
=A cos )
v=A cos ) ………… (7)
The minimum value of v is 0(as minimum value of Asin )= & maximum value is A. The maximum value of v is called velocity amplitude.
Acceleration in SHM:-
a= -A 2 sin ) …………. (8)
The minimum value of „a‟ is 0 & maximum value is A 2. The maximum valueof „a‟ is called acceleration amplitude.
Also, a= 2 x (from equation (5))
a – y
It is also the condition for SHM.
Time period in SHM:-
The time required for one complete oscillation is called the time period (T). It is related to the angular frequency( ) by.
T= ……………… (9)
Total energy= K.E+P.E
= A^2 ω^2 cos^2 ) + 2 sin^2 )
= 2 cos^2 ) + 2 sin^2 )
Total energy = 2
Total energy = A^2 ω^2
The total energy of an oscillatory system is constant.
Graphical relation between different characteristics in SHM.
Here the restoring force is -mgsinθ. So the restoring torque about the point of suspension “O” is
τ=-mg sinθ.
If the moment of inertia of the body about “OA” is “I”, the angular acceleration becomes,
α=τ/I
α= ……………….(1)
For very small angular displace “θ “, we assume that
Sin θ~θ.
So, α=-mglθ/I.
α=-(mg /I) θ………. (2)
Also α=d^2 θ/dt^2
Now we can write
d^2 θ/dt^2 + ( mg /I) θ =0……………..(3)
d^2 θ/dt^2 +ω^2 θ=0……………..(4)
Where, ω^2 = mg /I. And eqn(4) is the general equation of simple harmonic.
T=2π(I/mg )1/
T=2π( M(k^2 +L^2 )/Mg )1/2.
T=2π( (K^2 /l+l)/g)1/2……………………………(5).
Here + =L, Called as equivalent length of pendulum..
If a line which is drawn along the line joining the point of suspension & Centre of gravity by the distance “ k^2 /l”.we have
another Point on the line called centre of Oscillation is equivalent Length of pendulum.
So,the distance between centre of suspension & centre of Oscillation is equivalent length of pendulum .If these two points are interchanged then “time period” will be constant.
L.C CIRCUIT(NON MECHANICAL OSCILLATION ):-
In this region,it is combination “L” &”C” with the DC source through the key.If we Press the Key for a while then capacitor get charged & restores the charge as “+Q” and”-Q” with the potential “v=q/c” between the plates .When the switch is off the capacitor gets discharged.
As capacitor gets discharged, q also decreases. So, current at that situation is given by
I=dq/dt.
ω= ⁄√
T=2π√
Damped oscillation:-
For a free oscillation the energy remains constant. Hence oscillation continues indefinitely. However in real fact, the amplitude of the oscillatory system gradually decreases due to experiences of damping force like friction and resistance of the media.
The oscillators whose amplitude, in successive oscillations goes on decreasing due to the presence of resistive forces are called damped oscillators, and oscillation called damping oscillation.
The damping force always acts in a opposite directions to that of motion of oscillatory body and velocity dependent.
Fdam – v Fdam=-bv b= damping constant which is a positive quantity defined as damping force/velocity,
Fnet = Fres+ Fdam Fnet= -kx – bv
Fnet= -kx– b
M +kx+ b = 0
+2β +ω 02 x = 0 …………. (2)
Where β= is the damping co-efficient & ω 0 =√ is
called the natural frequency of oscillating body.
The above equation is second degree linear homogeneous equation.
The general solution of above equation is found out by assuming x(t), a function which is given by
x(t) = A
= A = x
= Aα^2 = α^2 x
Putting these values in equation
α^2 x + 2α^2 βx + ω 02 x = α^2 + 2α^2 β + ω 02 =0 ………… (3)
α = - β±√ ω 02 , is the general solution of above quadratic equation.
As we know,
x(t) =A 1 + A 2
x(t) = A 1 (^ √^ )^ + A 2 (^ √^ )
x(t) = (A 1 √^ + A 2 √^ ) … (4)
Depending upon the strength of damping force the quantity (β^2 - ω 02 ) can be positive /negative /zero giving rise to three different cases.
Case-1:- if β ω 02 underdamping (oscillatory)
Case-2:- if β ω 02 overdamping (non-oscillatory)
Mean life time: The time interval in which the oscillation falls to 1/e of its initial value is called mean life time of the oscillator. (τ)
1/e a= a e-βτm = ,
=
Velocity of underdamped oscillation:
X(t)=r cos(ω 1 t+ θ)
r[-βe-βt^ cos(ω 1 t+ θ)-e-βtω 1 sin( ω 1 t+ θ)
= v = -re-βt[βcos(ω 1 t+ θ)+ω 1 sin( ω 1 t+ θ)…(vi)
Now , x=0& t=0,
X(t)= re-βt^ cos(ω 1 t+ θ)
0 = re^0 cos(0+ θ) 0 = cosθ
Using the value of θ & t=0 in the equation (vii) we have
= -r ω 1 Where value of V 0 in ……………
Calculation of Energy(instantaneous):
K.E = mv^2
K.E = mv^2 [β^2 cos^2 (ω 1 t+ θ)+ ω 12 sin^2 (ω 1 t+ θ)+ βω 1 sin^2 (ω 1 t+ θ)]
Potential Enegy:
P.E= kx^2
= kr^2 cos^2 (ω 1 t+ θ)
Total Energy:
T.E=K.E+P.E
= [( mv^2 + kr^2 )cos^2 (ω 1 t+θ)+ mr^2 ω 12 sin^2 (ω 1 t+θ)
mv^2 βω 1 sin2(ω 1 t+ θ)]
Total average energy:
= mr^2 ω 02
=E 0 Where, E 0 =Total energy of free oscillation
The average energy decipated during one cycle
= Rate of energy
Case-II: (over damping oscillation)
Here β^2 ω 02
√ β^2 - ω 02 =+ve quantity
= α (say)
X (t) = e-βt^ (A 1 eαt^ +A 2 e-αt)…………(viii) Depending upon the relative values of α, β ,A1 , A 2 & initial position and velocity the oscillator comes back to equilibrium position.
The motion of simple pendulum in a highly viscous medium is an example of over damped oscillation.
Quality factor:
Q= π =.
Critical damping: β^2 = ω 02 The general solution of equation (ii) in this case, X(t) = (Ct+D) e-βt^ …………………………………(ix)
Here the displacement approaches to zero asymptotically for given value of initial position and velocity a critically damped oscillator approaches equilibrium position faster than other two cases. Example: The springs of automobiles or the springs of dead beat galvanometer.
