OCSILLATION
Types of Motion:-
(a) Periodic motion:- When a body or a moving particle repeats its motion along a definite path after regular intervals of time, its motion is said to be Periodic Motion and interval of time is called time or harmonic motion period (T). The path of periodic motion may be linear, circular, elliptical or any other curve.
(b) Oscillatory motion:-‘ To and Fro’ type of motion is called an Oscillatory Motion. It need not be periodic and need not have fixed extreme positions.The force/torque acting in oscillatory motion (directed towards equilibrium point) is called restoring force/torque.Periodic Motion
(c) Simple harmonic motion (SHM):- Simple harmonic motion is the motion in which the restoring force is proportional to displacement from the mean position and opposes its increase.
Simple harmonic motion (SHM):-
A particle is said to move in SHM, if its acceleration is proportional to the displacement and is always directed towards the mean position.
Conditions of Simple Harmonic Motion
Simple Harmonic Motion
For SHM is to occur, three conditions must be satisfied.
(a) There must be a position of stable equilibrium
At the stable equilibrium potential energy is minimum.
So, dU/dy= 0 and d2U/dy2> 0
(b) There must be no dissipation of energy
(c) The acceleration is proportional to the displacement and opposite in direction.
That is,a = -ω2y
Equation of SHM:-
(a) F = -ky (b) d2y/dt2 +ω2y = 0
Here ω = √k/m (k is force constant)
Displacement (y) :-
Displacement of a particle vibrating in SHM, at any instant , is defined as its distance from the mean position at that instant.
Simple Harmonic Motion
y = r sin (ωt+?)
Here ? is the phase and r is the radius of the circle.
Condition:
When, ? = 0, then, y = r sin ωt
and
When, ? = π/2, then,y = rcosωt
Amplitude (r):-
Amplitude of a particle, vibrating in SHM, is defined as its maximum displacement on either side of mean position.
As the extreme value of value of ωt = ± 1, thus, y = ±r
Velocity (V):-
V= dy/dt = rωcos(ωt+?)= vcos(ωt+?) = ω√r2-y2
Here v is the linear velocity of the particle.
Condition:-
When, y = 0, then, V = v = rω
and
When, y = ±r, then, V=0
A particle vibrating in SHM, passes with maximum velocity through the mean position and is at rest at the extreme positions.
y2/r2 + y2/ω2r2 = 1
Acceleration (a):
a = dV/dt = (-v2/r) sinωt = -ω2y
Condition:-
When, y = 0, then, a = 0
And,
When, y = ±r, then, a = ±ω2r
A particle vibrating in SHM, has zero acceleration while passing through mean position and has maximum acceleration while at extreme positions.
(i) Acceleration is directly proportional to y (displacement).
(ii) Acceleration is always directed towards the mean position.
Time period (T):-
It is the time taken by the particle to complete one vibration.
(a) T = 2π/ω
(b) T =2π√(displacement/acceleration)
(c) T = 2π√m/k
Frequency (f):-
It is the number of vibrations made by the body in one second.
(a) f=1/T
(b) f=1/2π√k/m
Angular frequency (ω):-
(a) ω = 2π/T
(b) ω =√(acceleration /displacement)
Relation betweenAngular frequency (ω) and Frequency (f):- ω = 2πf=√k/m
Phase:-
(a) Phase of a particle is defined as its state as regards its position and direction of motion.
(b) It is measured by the fraction of time period that has elapsed since the particle crossed its mean position, last, in the positive direction.
(c) Phase can also be measured in terms of the angle, expressed as a fraction of 2π radian, traversed by the radius vector of the circle of reference while the initial position of the radius vector is taken to be that which corresponds to the instant when the particle in SHM is about to cross mean position in positive direction.
Energy in SHM:-
(a) Kinetic Energy (Ek):-
Ek = ½ mω2(r2-y2) = ½ mω2r2cos2ωt
When, y = 0, then, (Ek)max = ½ mω2r2 (maximum)
And
When, y = ±r, then, (Ek)min =0 (minimum)
(b) Potential Energy (Ep):-
Ep = ½ mω2r2 = ½ mω2r2sin2ωt
(Ep)max = ½ mω2r2
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