IB DP Physics Topic 4: Waves: 4.1 Oscillations Study Notes

OSCILLATIONS

PERIODIC AND OSCILLATORY MOTION

PERIODIC MOTION

• Motion of moon around earth
• Motion of a piston in a cylinder
• Motion of a simple pendulum etc.

OSCILLATORY MOTION

• The motion of a ball in bowl

• The needle of a sewing machine
• Vibrations of prongs of tuning fork etc.

• All oscillatory motion are periodic but all periodic motion are not oscillatory motion.
• The oscillatory motion which can be expressed in terms of sine and cosine function, is said to be harmonic motion.

SIMPLE HARMONIC MOTION (S.H.M.)

• Motion of a body suspended by a spring
• Oscillations of simple pendulum

EQUATIONS OF S.H.M.

or,

or,

where C is called torsional rigidity

or, 

or, 

TERMS RELATED TO S.H.M.

• Amplitude : The maximum displacement of the oscillating particle on either side of its mean position is called its amplitude. It is denoted by A.
• Time period : The time taken by a oscillating particle to complete one oscillation is called its time period. It is denoted by T.
• Frequency : It is the number of oscillations completed in one second.
  It is denoted by υ.
  

• Angular frequency

• Phase : The parameter, by which the position of particle from its mean position is represented, is known as phase. The phase at any instant tells the state of position & direction of motion at that instant. The phase at time t = 0 is known as the initial phase or epoch (e).
• Total phase angle : The total angle (ωt + θ) is known as total phase angle.

CHARACTERISTICS OF S.H.M.

DISPLACEMENT
The displacement of a particle in S.H.M. is given by

or,

At y = 0 (at mean position),

or a = – ω2y

• Kinetic energy : A particle in S.H.M. possesses kinetic energy by virtue of its motion.

= \(K(t)=\frac{1}{2}mv^2 =\frac{1}{2}mw^2x_m^2sin^2(wt+\phi )\)

• Potential energy : A particle in S.H.M. possesses potential energy due to its displacement from the mean position.

= \(U(t)=\frac{1}{2}kx^2 =\frac{1}{2}kx^2cos^2(wt+\phi )\)

• Total mechanical energy

As time changes, the energy shifts between the two types, but the total is constant.

1. 1. Restoring force F = – Mω2x
   2. Kinetic energy = (1/2) Mω2(A2 – x2)
   3. Potential energy = 1/2 Mω2x2
   4. Total energy of SHM = 1/2 Mω2A2

1. 5. The kinetic and potential energy of SHM varies sinusoidally with a frequency twice that of SHM.
   6. Total energy \(\frac{1}{2}m\omega^2 A^2 = 2\pi^2 m A^2 n^2\)

1. 7. where ω is constant

    

1. 8. Geometrically the projection of the body undergoing uniform circular motion on the diameter of the circle is SHM.
   9. In a non-inertial frame.
      
      

SOME SYSTEMS EXECUTING S.H.M.

• When two springs having force constants k1 and k2 connected in parallel, then

• When two springs of force constants k1 and k2 are connected in series, then

Hence

• If two mass M1 and M2 are connected at the two ends of the spring, then their period of oscillation is given by

T = 2π[μ/k)]1/2 where is the reduced mass.

• When the length of spring increases, spring constant decreases. If the length of spring becomes n times, its spring constant becomes times and therefore time period will be increased by times.
• If we divide the spring into n equal parts, the spring constant of each part becomes n k. Hence time period when the same mass is suspended from each part is:

  …(i)

 …(ii)

Step 1 : First, find the equilibrium position. Equilibrium position will be one for which and

Find the restoring torque acting on the body (for rotational)

Step 3 : Since

∴ Use  … (i) for translational

      … (ii) for rotational

Step 4 : (for translational)

  (for rotational) where, I = moment of inertia

Incorrect. The time taken to cover half the amplitude form equilibrium position is .

Correct. The actual time taken is .

Rectangular block floating in a liquid,

where d = density of liquid, d′ = density of block, h = height of block

Time period,

CASE 6 – If a tunnel is dig in the earth diametrically or along a chord then time period, for a particle released in the tunnel.

CASE 7 – The time period of a ball oscillating in the neck of a chamber

CASE 8 – If a dipole of dipole moment p is suspended in a uniform electric field E then time period of oscillation

1. 1. In S.H.M. the phase relationship between displacement, velocity and acceleration, is as follows :
      1. The velocity is leading the displacement by a phase radian
      2. The acceleration is leading the displacement by a phase π radian
      3. The acceleration is leading the velocity by a phase radian.
      4. When , then velocity V = 0.86Vmax.
      5. When V = Vmax/2, the displacement x = 0.87A.
      6. When , the kinetic energy of S.H.M. is 75% of the total energy and potential energy 25% of the total energy.
      7. When the kinetic energy of S.H.M. is 50% of the total energy, the displacement is 71% of the amplitude.
   2. The time period of a simple pendulum of length l which is comparable with radius of earth.

1. 1. 1. When l << R, then
      2. When l = R, we find
      3. When l = , then

= 84.6 minutes. Thus maximum of T is 84.6 minutes.

1. 4. 4. Under weightlessness or in the freely falling lift
         
         This means, the pendulum does not oscillate at all.
   5. Time period of a simple pendulum in a train accelerating or retarding at the rate a is given by

1. 4. If a simple pendulum whose bob is of density do is made to oscillate in a liquid of density d, then its time period of vibration in liquid will increase and is given by

(where d0 > d)

1. 5. The time period of a simple pendulum in a vehicle moving along a circular path of radius r and with constant velocity V is given by,
   6. If T1 and T2 are the time periods of a body oscillating under the restoring force F1 and F2 then the time period of the body under the influence of the resultant force will be
   7. (a) The percentage change in time period of simple pendulum when its length changes is
      1. 

(b) The percentage change in time period of simple pendulum when g changes but l remains constant is

(c) The percentage change in time period of simple pendulum when both l and g change is

1. 8. If a wire of length l, area of cross-section A, Young’s modulus Y is stretched by suspending a mass m, then the mass can oscillate with time period
      
   9. If a simple pendulum is suspended from the roof of compartment of a train moving down an inclined plane of inclination θ, then the time period of oscillations

1. 10. If a ball of radius r oscillates in a bowl of radius R, then its time period is given by :
   11. If a disc of radius r oscillates about a point at its rim, then its time period is given by:

1. 12. The graph between the length of a simple pendulum and its time period is a parabola.
   13. The graph between the length of a simple pendulum and the square of its time period is a straight line.
   14. The graph between l & T and between l & T2 intersect at T = 1 second.
   15. The time period of the mass attached to spring does not change with the change in acceleration due to gravity.
   16. If the mass m attached to a spring oscillates in a non-viscous liquid density σ, then its time period is given by

1. 17. The length of second pendulum (T = 2 sec) is 99 cm

PHYSICAL PENDULUM

and

CONICAL PENDULUM

If l is the length of the pendulum and the string makes an angle θ with vertical then time period,

TORSIONAL PENDULUM

1. 1. The displacement, velocity and acceleration of S.H.M. vary simple harmonically with the same time period and frequency.
   2. The kinetic energy and potential energy vary periodically but not simple harmonically. The time period of kinetic energy or potential energy is half that of displacement, velocity and acceleration.
   3. The graph between displacement, velocity or acceleration and t is a sine curve. But the graph between P.E. or K.E. of S.H.M. and time t is parabola.

1. 4. If the bob of simple pendulum is -vely charged and a +vely charged plate is placed below it, then the effective acceleration on bob increases and consequently time period decreases.

Time period,

1. 5. If the bob of a simple pendulum is -vely charged and is made to oscillate above the -vely charged plate, then the effective acceleration on bob decreases and the time period increases.
      

1. 6. A pendulum clock slows down in summer and goes faster in winter.
   7. Potential energy of a particle executing S.H.M. is equal to average force × displacement.

i.e., .

1. 8. If the total energy of a particle executing S.H.M. is E, then its potential energy at displacement x is

and kinetic energy

FREE, DAMPED, FORCED OSCILLATIONS AND RESONANCE

FREE OSCILLATION

= Frestoring force = –kx, where k is constant.

, where ω0 is natural frequency of body.

The time period is

DAMPED OSCILLATION

 …(i)

where 2 C = b/m (C is damping constant) & , the natural frequency of oscillating particle i.e., its frequency in absence of damping.

sin (ωt + φ)  …(ii)

and (relaxation time)

It is clear from the fig & eqn. (ii) that the amplitude of damped harmonic oscillator decreases with time. In this case, the motion does not repeat itself & is not periodic in usual sense of term. However it has still a time period, , which is the time interval between its successive passage in same direction passing the equilibrium point.

FORCED OSCILLATION AND RESONANCE

 …… (i)

⇒ 

or   …… (ii)

where

where & p is the frequency of external periodic force.