INTRODUCTION:
Periodic motion is the type of motion that repeats itself after regular intervals of time
over and over again.
The oscillatory or vibratory motion is the type of motion in which an object moves to and
fro or back and forth in a definite interval of time in a repetitive manner about a fixed
point.
A specific kind of oscillatory motion is simple harmonic motion, which includes the
following:
1. the particle moves are restricted to a single dimension
2. Whenever Fnet=0, the particle oscillates back and forth around a constant mean
location.
3. The equilibrium position is always the direction of the net force acting on the particle.
4. The particle’s displacement from the equilibrium location at that instant is always
inversely proportional to the size of the net force.
1.1 Some Important Terms:
The amplitude of a particle executing S.H.M. is its maximum displacement on either
side of the equilibrium position. It is represented by A.
The time period of a particle executing S.H.M. is the time taken to complete one cycle
and is represented by T.
The frequency of a particle executing S.H.M. is denoted by v and is the same as the
number of oscillations completed in one second.
The phase of particles executing S.H.M. at any instant is its state with respect to the
direction of motion and its position at that particular instant. Phase=(ωt+ϕ)
1.2 Important Relations:
CBSE Class 11 Physics Chapter 14 Notes explain the following in detail:
1. Position– When the equilibrium position is at origin, the position depends on time in
general as x (t)=sin (ωt+ϕ)
At the equilibrium position, x = 0 while at the extremes, x=+a,−a
2. Velocity-
v(t)=Aωcos(ωt +ϕ) at any instant t.
v(x)=±ω√(A2−x2) at any position x.
Since the particle is at rest at the extremes, Velocity has minimum magnitude here i.e.
v=0 at the extreme position.
Velocity has maximum magnitude at the equilibrium position i.e.|v|max=ωA here.
3. Acceleration-
If the spring is suspended vertically from a fixed point while being carried by its opposite end as
shown, the block will oscillate along the vertical line.
Mean position: Via d=(mg)/k, the spring in elongated
Time period: T=2π√(m/k)
c) Combination of springs:
Springs in series:
Take into account two springs with the corresponding force constants K1 and K2, connected in
series as indicated. They are comparable to one spring with force constant K, denoted by
1/K=1/K1+1/K2
⇒K=K1K2/K1+K2
Springs in parallel:
The e ective spring constant for a parallel combination is K=K1+ K2
2.2 Oscillation of a Cylinder Floating in a Liquid:
Suppose the total length of the cylinder is L and a cylinder of mass m and density d be floating
on the surface of a liquid of density ρ.
Mean position: It is where the cylinder is immersed up to ℓ=(Ld)/ρ
Time period: T=2π√[(Ld)(ρg)] =2π√(ℓ/g)
2.3 Liquid Oscillating in a U–Tube:
In a U-tube of the area of cross-section A, suppose a liquid column of mass m and density ρ.
Mean position: It is when the height of the liquid is the same in both limbs.
Time period: T=2π√[m/(2Aρg)]= 2π√L/(2g)
2.4 Body Oscillation in the Tunnel Along any Chord of the Earth:
Mean position: It is at the centre of the chord
Time period: T=2π√(Rg) = 84.6 minutes where the radius of the earth is R.
2.5 Angular Oscillations:
When a centre or particle of mass of a body oscillates on a small arc of the circular path,
instead of straight-line motion, then it is called an angular S.H.M.
For angular S.H.M., torque is:
τ=kθ
where θ is the angular displacement and k is a constant.
⇒ Iα=−kθ
where α is the angular acceleration and I is the moment of inertia.
