Kerala Plus One Physics Notes Chapter 14 Oscillations
Any motion that repeats itself over and over again at regular intervals of time is called periodic or harmonic motion.
eg., the revolution of earth around sun, the motion of hands of a clock, heart beats of a human being etc.
Oscillatory motion (Vibratory motion):
It is that motion in which a body moves back and forth repeatedly about a fixed point (called mean position), in a definite interval of time.eg., motion of pendulum of a wall clock, vibration of a string of a guitar etc.
Simple Harmonic Motion (S.H.M):
A particle is said to execute SHM if it moves to and fro about a mean position under the action of a restoring force which is directly proportional to its displacement from the mean position and is always directed towards the mean position.
Consider a particle executing SHM along the x-axis between points A and B with 0 as the mean position. Let at instant t the particle be at P, where the displacement from mean position is x.
The restoring force acting on the particle at that instant is. F =-kx . where k is called the force constant. The negative sign shows that the force is opposing the motion. The above relation is called force law of SHM. Thus a system will execute SHM if it obeys the above relation.
By Newton’s second law,
This is the differential equation of SHM.
A possible solution to the above equation may be of the form x = A cos (ωt + φ)
Some important terms in SHM:
- Harmonic oscillator: A particle executing harmonic motion.
- Displacement (x): The distance of oscillating particle from its mean position at any instant is called its displacement.
- Amplitude (A): It is the maximum displacement of the oscillating particle on either side of its mean position. Thus xmax– ±A
- Time period (T): It is the time taken by a particle to complete one oscillation.
- Frequency: It is defined as the number of oscillations completed per unit time by a particle. Frequency is equal to the reciprocal of time period, ie., f = 1/T
SI unit is s-1 = Hz (hertz) = cps
- Angular frequency (ω): Angular frequency is equal to the product of frequency of the body and 2n.
Relation between angular frequency, time period and frequency in SHM:
We see that the in SHM the displacement x(t) must return to its initial value after one period T of the motion, ie., x(t) must be equal to x(t+T). Taking, we write
A cos ωt = A cos ω(t + T)
As the cosine function repeats itself when the phase is increased by 2π, we get
Simple Harmone Motion and Uniform Circular Motion:
Fig(1) shows a reference particle P’ moving in uniform circular motion with angular velocity co. The radius A of the circle is the magnitude of the particle’s position vector. ωt+φ is the angular position of the particle at any time t, where φ is the angular position at t = 0.
This is similar to the equation of SHM we have seen before. So, if the reference particle P’ is moving in uniform circular motion, its projection particle P moves in SHM along a diameter of the circle.
Thus simple harmonic motion is the projection of uniform circular motion on a
diameter of the circle in which the circular motion occurs velocity.
Fig (2) shows the velocity v of the reference particle. From the equation in circular motion (v =ωt), the magnitude of the velocity vector is ωA (radius = A)
The minus sign appears because the velocity of P’ is directed towards left, ie., in the negative x-direction.
Hence velocity of particle in SHM is zero at both extreme positions.
Fig(3) shows the radial acceleration a of the reference particle. From the equation
ar = ω2 r the magnitude of the radial acceleration vector is ω2 A and from the triangle POP’ we get,
a(t)=-ω2 A cos ( ωt + φ) = -ω2x
At mean position, x = 0, a = -ω2(0) = 0
At extreme positions, x = A, a = -ω2 A
This is, the maximum value of acceleration which a particle in SHM can have.
From the above relation of acceleration we get yet another expression for time period,
Phase relation between displacement, velocity and acceleration
If the distance y of a point moving on a straight line measured from a fixed origin on it and velocity v are connected by the relation 4v2 = 25 – y2, then show that the motion is simple harmonic and find its time period.
Comparing the above two equations, we find that the given equation represents SHM of amplitude A = 5 and ω = 1/2 rad s-1.
Energy in Simple harmonic motion:
A simple harmonically moving particles possesses both potential energy and kinetic
energy. Potential energy is due its displacement against restoring force. Kinetic energy due to its motion.Total energy of the S.H.M. is the sum of the kinetic energy and potential energy. Total energy remains a constant throughout its motion.
Let m be the mass of the particle executing SHM. Let v be the velocity at any instant,
Potential energy is the work required to take a particle against the restoring force.
Let a particle be displaced through a distance x from the mean position. Then restoring force, F = -kx, where k is the force constant. Now if we displace the particle further through a distance dx,
Small work done, dw = -Fdx = kx dx
Total work done from O to x
Some System Executing S.H.M
Oscillations due to a Spring:
The simplest example is the small oscillations of a block mass ‘m’ fixed to a wall. The block moves horizontally on a fixed frictionless surface. When the block is pulled sideways slightly and released it executes simple harmonic motion in the horizontal direction.
- The time period will be small and the frequency will be large if the spring is stiff (high k) and attached body is light (small m).
- Spring constant k of a spring is the force required to produce an elongation of 1m in the spring.
- Springs connected in parallel: If two springs having spring constants k1 and k2 are connected in parallel supporting a mass m, then the spring constant of the combination is k = k1 +k2
- Springs connected in series: If two springs having spring constants k1 and k2 are connected in series supporting a mass m then spring constant.
The pan attached to a spring balance has a mass of 1 kg. A weight of 2 kg when placed on the pan stretches the spring by 10 cm. What is the frequency with which the empty pan will oscillate?
An ideal simple pendulum consists of a heavy point mass suspended by a weightless, inelastic and flexible string from a rigid support about which it is free to oscillate.
Tension,T cancels the radial component mg cosθ. The tangential component mg sinθ provides the restoring force to bring the bob back to the equilibrium position (θ = 0). The restoring force, F =-mg sin e
F = -mg θ (as θ is small, sin θ=0)
As the restoring force is proportional to the angular displacement the pendulum
The restoring torque, τ =FxI ;τ = -mgl θ We know that, τ = lα
(I = rotational inertia about point of suspension and α= angular acceleration about that point)
Thus the time period of a simple pendulum depends on its length and acceleration due to gravity and is independent of the mass of the bob.
Undamped simple harmonic oscillation:
When a simple harmonic system oscillates with a constant amplitude which does not change with time, its oscillations are called undamped simple harmonic oscillations.
Damped simple harmonic oscillation:
When a simple harmonic system oscillates with a decreasing amplitude with time, its oscillations are called damped simple harmonic oscillations, eg., the oscillations of a swing in air, the oscillations of the bob of a pendulum in a fluid.
Forced oscillations and Resonance:
A swing without anyone pushing is oscillating freely. It will slowly decrease amplitude as free oscillations are practically impossible, Now, if we push the swing from behind, the swinging can be maintained. But now the swinging is not free it is forced, or driven. In this case, we have to consider two angular frequencies.
- The natural frequency ω of the system.
- The angular frequency ωd of the driving force.
Thus, forced oscillations is when a body oscillates with the help of an external periodic force with a frequency different from the natural frequency of the body.
In the above case of forced oscillations, if we push the swing with an angular frequency equal to the natural frequency of the system ie., (ωd=ω, then we get maximum velocity amplitude. This condition is called resonance
Thus if you push a swing at its natural angular frequency ω the displacement and
velocity amplitudes will increase to large values, Children learn this easily by trial and error. If you push at other angular frequencies, the displacement and velocity amplitudes will be smaller.
- A vibrating tuning fork when placed near the mouth of a particular length of air column produces a loud sound due to resonance.
- Army while crossing a suspension bridge breaks its steps (not in unison). This is to avoid resonance between bridge and the impressed force of their feet. Otherwise the bridge may collapse due to large amplitude in case of resonance.
Did You Know? The Tacoma Bridge in Washington was opened is 1940. Four months after the opening winds produced a fluctuating resultant force in resonance with the natural frequency of the bridge. This increased the amplitude of oscillation steadily and finally the bridge was destroyed.