Where l = effective length of the pendulum and g = acceleration due to gravity. The time period of the simple pendulum is given by : How the different physical quantities (e.g., displacement, velocity, acceleration, kinetic energy etc) vary with time or displacement are listed ahead in tabular form.Ī simple pendulum consists of a heavy point mass suspended from a rigid support by means of an elastic inextensible string. In general, inertia factor = m, (mass of the particle) When a particle of mass m executes SHM with a constant angular frequency (I), then time period of oscillation The frequency of total energy of particles executing SHM is zero as total energy in SHM remains constant at all positions. The frequency of kinetic energy or potential energy of a particle executing SHM is double than that of the frequency in SHM. = 1 / 2 mω 2 (A 2 – x 2) + 1 / 2 mω 2 x 2Ĭhanges of kinetic and potential energies during oscillations. Total energy can be obtained by adding potential and kinetic energies.

The potential energy of the particle is U = 1 / 2 mω 2 x 2įrom this expression we can see that, the potential energy has a minimum value at the centre (x = 0) and increases as the particle approaches either extreme of the oscillation (x ± A). The kinetic energy of the particle is K = 1 / 2 mω 2 (A 2 – x 2)įrom this expression we can see that, the kinetic energy is maximum at the centre (x = 0) and zero at the extremes of oscillation (x ± A). In SHM, the force is proportional and opposite to the displacement. Where, ω = √k / m and k = mω 2 is a constant and sometimes it is called the elastic constant. We know that, the acceleration of body in SlIM is α = -ω 2 x The graph between velocity and displacement for a particle executing SHM is elliptical. (iii) Instantaneous acceleration and instantaneous displacement (ii) Instantaneous velocity and instantaneous acceleration (i) Instantaneous displacement and instantaneous velocity Note: The acceleration is maximum at a place where the velocity is minimum and vice – versa.įor a particle executing SlIM. Negative sign indicates that the direction of acceleration is opposite to the direction in which displacement increases, i.e., towards mean position.Īt mean position y = 0 and acceleration is also zero.Īt extreme position y = a and acceleration is maximum (iii) Acceleration of a particle executing SHM at any instant is given by (ii) Velocity of a particle executing SHM at any instant is given by (i) Displacement in SHM at any instant is given by Simple harmonic motion is defined as the projection of a uniform circular motion on any diameter of a circle of reference. Phase A physical quantity which express the position and direction of motion of an oscillating particle, is called phase.Amplitude The maximum displacement in any direction from mean position is called amplitude.

Displacement A physical quantity which changes uniformly with time in a periodic motion.Angular Frequency The product of frequency with factor 2π is called angular frequency.Frequency The number of oscillations completed by the body in one second is called frequency.Time Period Time taken by the body to complete one oscillation is known as time period.It can be expressed as y = a sin ωt + b sin 2ωt The oscillation which can be expressed in terms of single harmonic function, i.e., sine or cosine function, is called harmonic oscillation.Ī harmonic oscillation of constant amplitude and of single frequency under a restoring force whose magnitude is proportional to the displacement and always acts towards mean Position is called Simple Harmonic Motion (SHM).Ī simple harmonic oscillation can be expressed asĪ non-harmonic oscillation is a combination of two or more than two harmonic oscillations. Note Every oscillatory motion is periodic motion but every periodic motion is not oscillatory motion. e.g., orbital motion of the earth around the sun, motion of arms of a clock, motion of a simple pendulum etc.Ī periodic motion taking place to and fro or back and forth about a fixed point, is called oscillatory motion, e.g., motion of a simple pendulum, motion of a loaded spring etc. A body is said to execute simple harmonic motion when it experiences a restoring force which is proportional to its displacement from equilibrium position and is directed to the equilibrium position.Ī motion which repeats itself identically after a fixed interval of time is called periodic motion.