Definitions [31]
Define the term oscillation.
Oscillation refers to a complete 'to and fro' motion of the body along its course.
A mathematical function which represents a periodic motion is called a periodic function.
Any motion which repeats itself after a definite interval of time is called periodic motion.
A fixed time period after which a motion repeats itself is called the time period.
A particular type of periodic motion in which a particle moves to and fro under the influence of restoring force about a mean position, where the acceleration is directly proportional to displacement from the mean position and its direction is always towards the mean position, is called Linear Simple Harmonic Motion.
Define linear simple harmonic motion.
Linear simple harmonic motion (S.H.M.) is defined as the linear periodic motion of a body, in which the restoring force (or acceleration) is always directed towards the mean position and its magnitude is directly proportional to the displacement from the mean position.
Define linear S.H.M.
Linear S.H.M. is defined as the linear periodic motion of a body in which the restoring force (or acceleration) is always directed towards its mean position and its magnitude is directly proportional to the displacement from the mean position.
Consider a particle ‘P’ moving along the circumference of a circle of radius 'a' and centre O, with uniform angular speed of 'ω' in anticlockwise direction as shown.
Particle P along the circumference of the circle has its projection particle on diameter AB at point M.
The distance an object in SHM travels from its initial position as a function of time is called displacement.
The rate of change of displacement with respect to time at any instant by a particle executing SHM is called velocity.
The rate of change of velocity with respect to time at any instant of a particle executing SHM is called acceleration.
The time taken by the particle performing SHM to complete one oscillation is called period of SHM.
The maximum displacement of a particle performing SHM from its mean position is called amplitude.
A combination in which two or more springs are connected end to end such that the effective spring constant is given by the reciprocal sum of individual spring constants is called series combination of springs.
A combination in which two or more springs are connected side by side such that the effective spring constant is the sum of individual spring constants is called parallel combination of springs.
A way to analyse simple harmonic motion (SHM) by using a circle as reference is called the reference circle method.
The state of oscillation of a particle performing SHM, represented by the angular displacement θ, is called phase in SHM.
The initial phase of a particle i.e., the phase at time t=0t=0 is called epoch.
The pictorial representation of variation of displacement, velocity, and acceleration of a particle performing SHM with respect to time (or ωtωt) is called graphical representation of SHM.
The energy possessed by a particle performing SHM due to its motion, which is maximum at the mean position and minimum (zero) at the extreme position, is called kinetic energy.
The sum of kinetic energy and potential energy of a particle performing SHM is called total energy.
The energy possessed by a particle performing SHM due to its displacement from the mean position, which is maximum at the extreme position and minimum at the mean position, is called potential energy.
Define an ideal simple pendulum.
An ideal simple pendulum consists a point mass suspended from a perfectly rigid support by weightless, inextensible and perfectly flexible fibre.
An ideal simple pendulum is a heavy particle suspended by a massless, inextensible, flexible string from a rigid support.
Define second’s pendulum.
A simple pendulum whose period of oscillation is exactly two seconds is called a second’s pendulum.
A heavy but small sized metallic bob suspended by a light, inextensible and flexible string, which performs oscillatory motion, is called a simple pendulum.
A simple pendulum whose period is two seconds is called a second's pendulum.
A type of oscillatory motion where an object undergoes periodic angular displacement about an equilibrium position due to a restoring torque, which is the rotational analogue of linear SHM, is called angular SHM.
The oscillations of a body whose amplitude goes on decreasing with time due to the presence of dissipative forces are called damped oscillations.
The oscillations of a body whose amplitude remains the same throughout the time are called undamped oscillations.
The oscillation of a body under the influence of an external periodic force is called forced oscillation.
The oscillations in which a body or system oscillates with its own natural frequency without being acted upon by an external force are called free oscillations.
The phenomenon that occurs when an external periodic force matches the natural frequency of a system, leading to maximum amplitude of oscillation, is called resonance.
Formulae [23]
F(t) = F(t + T)
where T is the time period of the periodic function.
F = −kx, a = −\[\frac {kx}{m}\]
\[\frac{d^2x}{dt^2}+\frac{k}{m}x\] = 0 OR \[\frac{d^2x}{dt^2}+\omega^2x\] = 0
| Property | Series Combination | Parallel Combination |
|---|---|---|
| Effective spring constant (two springs) | \[\frac{1}{k}=\frac{1}{k_1}+\frac{1}{k_2}\] | k = k1 + k2 |
| Effective spring constant (n identical springs) | keq = \[\frac{k}{2}\] (for 2 springs) keq = \[\frac{k}{n}\] (for n springs) |
keq = nk |
| Time period of oscillations | T = 2π \[\sqrt{\frac{m(k_1 + k_2)}{k_1 k_2}}\] | T = 2π \[\sqrt{\frac{m}{k_1 + k_2}}\] |
θ = (ωt + ϕ)
where ω = angular frequency, t = time, ϕ = initial phase (epoch).
vy = rω cos(ωt + ϕ), v = rω
U = \[\frac {1}{2}\]kx2 = \[\frac {1}{2}\]kA2 cos2 (ωt + Φ)
E = U + K = \[\frac {1}{2}\]kA2 = \[\frac {1}{2}\]mω2 A2
K = \[\frac {1}{2}\]mv2 = \[\frac {1}{2}\]mω2 A2 sin2(ωt + Φ) = \[\frac {1}{2}\]KA2 sin2 (ωt + Φ) = \[\frac {1}{2}\]k(A2 - x2)
T = 2π\[\sqrt {\frac {l}{g}}\]
n = \[\frac {1}{2π}\]\[\sqrt {\frac {g}{l}}\]
T = 2π\[\sqrt {\frac {L_s}{g}\] = 2 seconds
\[I\frac{d^2\theta}{dt^2}+C\theta=0\]
ω = \[\sqrt {\frac {k}{m}\]
\[\omega^{\prime}=\sqrt{\frac{k}{m}-\left(\frac{b}{2m}\right)^2}\]
\[\frac{d^2x}{dt^2}+\omega^2x=0\]
\[m\frac{d^{2}x}{dt^{2}}+b\frac{dx}{dt}+kx\] = 0 & emsp; (b = damping constant)
x = A sin(ωt + α)
x = Ae−bt/2m sin(ω′t + ϕ)
A = Ae−bt/2m
\[\frac{d^2x}{dt^2}+b\frac{dx}{dt}+kx=F_0\cos\omega_dt\]
x = A cos(ωdt + ϕ)
A = \[\frac{F_0}{\sqrt{m^2(\omega^2-\omega_d^2)^2+\omega_d^2b^2}}\]
ω = ω0
where ω = driving angular frequency and ω0 = natural angular frequency of the system.
At resonance: A = \[\frac {F_0}{ω_0b}\]
Key Points
(a) Particle starts from mean position (towards positive direction):
| Quantity | Expression | Range |
|---|---|---|
| Displacement | x = A sin(ωt) | −A to + A |
| Velocity | v = Aω cos(ωt) | −Aω to + Aω |
| Acceleration | a = −Aω2 sin(ωt) | −Aω2 to + Aω2 |
(b) Particle starts from positive extreme position:
| Quantity | Expression | Range |
|---|---|---|
| Displacement | x = A cos(ωt) | −A to + A |
| Velocity | v = −Aω sin(ωt) | −Aω to + Aω |
| Acceleration | a = −Aω2 cos(ωt) | −Aω2 to + Aω2 |
- When a particle is subjected simultaneously to two SHMs having the same period and along the same path, the resultant motion is also SHM along the same path.
- The resultant displacement of the particle at any instant is equal to the vector sum of its individual displacements due to both the SHMs at that instant.
- The total mechanical energy of a harmonic oscillation is independent of time, as expected for motion under any conservative force.
- Both kinetic and potential energies peak twice during each period of SHM.
- Period of kinetic energy and potential energy = \[\frac {T}{2}\].
Concepts [16]
- Oscillations
- Explanation of Periodic Motion
- Linear Simple Harmonic Motion (S.H.M.)
- Differential Equation of Linear S.H.M.
- Acceleration (a), Velocity (v) and Displacement (x) of S.H.M.
- Amplitude (A), Period (T) and Frequency (N) of S.H.M.
- Combination of Springs
- Reference Circle Method
- Phase in S.H.M.
- Graphical Representation of S.H.M.
- Composition of Two S.H.M.’S Having Same Period and Along Same Line
- The Energy of a Particle Performing S.H.M.
- Simple Pendulum
- Angular S.H.M. and It's Differential Equation
- Damped Oscillations
- Free Oscillations, Forced Oscillations and Resonance Oscillations
