SIMPLE HARMONIC MOTION Chapter 1 Physics Paper B BSc. I

Motion of a body PERIODIC MOTION- The motion which repeats itself at a regular intervals of time is known as Periodic Motion. Examples are: a) Revolution of earth around sun b) The rotation of earth about its polar axis c) The motion of simple pendulum OSCILLATORY OR VIBRATORY MOTION- The periodic motion and to and fro motion of a particle or a body about a fixed point is called the oscillatory or vibratory motion. Examples are: a) Motion of bob of a simple pendulum b) Motion of a loaded spring c) Motion of the liquid contained in U-tube All oscillatory motions are periodic but all periodic motions are not oscillatory.

Simple Harmonic Motion (S.H.M) DEFINITION S.H.M is a motion in which restoring force is 1. directly proportional to the displacement of the particle from the mean or equilibrium position. 2. always directed towards the mean position. i.e. F  y F = -ky where k is the spring or force constant. The negative sign shows that the restoring force is always directed towards the mean position.

Example1 Mass-Spring System a -is the acceleration a aa a Equilibrium position

Example2 a a a a Equilibrium position Simple Pendulum

Characteristics of S.H.M Equilibrium: The position at which no net force acts on the particle. Displacement: The distance of the particle from its equilibrium position. Usually denoted as y(t) with y=0 as the equilibrium position. The displacement of the particle at any instant of time is given as Amplitude: The maximum value of the displacement without regard to sign. Denoted as r or A.

Characteristics of S.H.M Velocity: Rate of change of displacement w.r.t time. Acceleration: Rate of change of velocity w.r.t time. Phase: It is expressed in terms of angle swept by the radius vector of the particle since it crossed its mean position.

Time Period and Frequency of wave Time Period T of a wave is the amount of time it takes to go through 1 cycle. Frequency f is the number of cycles per second.  the unit of a cycle-per-second is commonly referred to as a hertz (Hz), after Heinrich Hertz ( ), who discovered radio waves. Frequency and Time period are related as follows: Since a cycle is 2  radians, the relationship between frequency and angular frequency is: T

Displacement-Time Graph y = rsin(  t) t 0 r -r y

Velocity-Time Graph v = r  cos(  t) t 0 rr  r 

Acceleration-Time Graph t 0 a rr rr a =  r   sin(  t)

Phase Difference o Fig.1 shows two waves having phase difference of  or 180 o. o Fig. 2 shows two waves having phase difference of  /2 or 90 o. o Fig.3 shows two waves having phase difference of  /4 or 45 o.

Differential Equation of Simple Harmonic Motion When an oscillator is displaced from its mean position a restoring force is developed in the system. This force tries to restore the mean position of the oscillator. (1) where k is the spring or force constant. From Newton’s second law of motion, (2) Comparing (1) and (2) we get We can guess a solution of this equation as y = rsin(  t+  ) Or y = rcos(  t+  ) where  is the phase angle.

Energy of a Simple Harmonic Oscillator A particle executing S.H.M possesses two types of energies: a) Potential Energy: Due to displacement of the particle from mean position. b) Kinetic energy: Due to velocity of the particle.

Total Energy Total energy of the particle executing S.H.M is sum of kinetic energy and potential energy of the particle. Total energy is independent of time and is conserved.

Simple Pendulum mgsin  mgcos   A Simple Pendulum is a heavy bob suspended from a rigid support by a weightless, inextensible and heavy string. Component mgcos θ balances tension T.

Simple Pendulum Where T is time period of pendulum.

Compound Pendulum Definition: A rigid body capable of oscillating freely in a vertical plane about a horizontal axis passing through it. If we substitute torque Restoring force = -mglsin θ Assuming  to be very small, sin    which is angular equivalent of Where I is moment of inertia of body and α is angular acceleration.

Compound Pendulum Time Period is where I is the moment of inertia of the pendulum. Centre of suspension and centre of oscillation are interchangeable.

Torsional Pendulum If the disk is rotated through an angle (in either direction) of , the restoring torque is given by the equation: Comparing with F = -kx which gives Time period of oscillations

In mechanical oscillator we have force equation and it becomes voltage equation in electrical oscillator. A circuit containing inductance(L) and capacitance(C) known as tank circuit which serves as an electrical oscillator. Differential equation for Electrical Oscillator where Solution of this equation is Simple harmonic Oscillations in an Electrical Oscillator

Energy of Electrical Oscillator In an electrical oscillator we have two types of energies: Electrical energy stored in capacitor Magnetic energy stored in inductor Total energy of electrical oscillator at any instant of time is

Comparison of Mechanical and Electrical Oscillator ParameterMechanical OscillatorElectrical Oscillator Equation of Motion EnergyTotal Mechanical energyTotal Electrical Energy Solution y = rsin(  t+  ) (or a cosine function) q = q 0 sin(  t+  ) (or a cosine function) InertiaMass mInductance L ElasticityStiffness k1/C What Oscillates?Displacement(y), Velocity(dy/dt), Acceleration(d 2 y/dt2) Charge(q), current(dq/dt), dI/dt Driving AgentForceInduced Voltage Frequency

Simple Harmonic Motion is the projection of Uniform Circular Motion

Lissajous Figure components in phase

Lissajous Figure components out of phase

Lissajous Figure x 90 o ahead of y

Lissajous Figure x 90 o behind y