SIMPLE HARMONIC MOTION Chapter 1 Physics Paper B BSc. I.

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 sunb) The rotation of earth about its polar axisc) 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 pendulumb) Motion of a loaded springc) 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 is1. 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 = -kywhere 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 accelerationa a a a

Equilibrium position

Example2

aa

aa

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 (1847-1894), 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

t

Displacement-Time Graph

y = rsin( wt)

t0

r

-r

y

Velocity-Time Graphv = rwcos(wt)

t0

rw

- rw

Acceleration-Time Graph

t0

a

rw2

-rw2

a = - rw2sin(t)

Phase Difference

o Fig.1 shows two waves having phase difference of or 180o .

o Fig. 2 shows two waves having phase difference of /2 or 90o.

o Fig.3 shows two waves having phase difference of /4 or 45o.

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 EnergyTotal 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

mgsinq

mgcosq

q

kymg

mg

sin

Force Restoringsin

A Simple Pendulum is a heavy bob suspended froma rigid support by a weightless, inextensible and heavy string.

Component mgcosθ balances tension T.

Simple Pendulum

k

mT

g

l

k

m

klmg

smallif

Lkmg

AmplitudeLsL

s

R

s

spring

2

,sin

sin

g

lTpendulum 2

Where T is time period of pendulum.

Compound PendulumDefinition: A rigid body capable of oscillating freely in a vertical plane about a horizontal axis passing through it .

If we substitute torque

Restoring force = -mglsinθAssuming to be very small,

sin

which is angular equivalent of

Where I is moment of inertia of body andα 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

PendulumIf the disk is rotated throughan angle (in either direction)of , the restoring torque isgiven 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

Parameter Mechanical Oscillator

Electrical Oscillator

Equation of Motion

Energy Total Mechanical energy

Total Electrical Energy

Solution y = rsin(t+) (or a cosine function)

q = q0 sin(t+) (or a cosine function)

Inertia Mass m Inductance L

Elasticity Stiffness k 1/C

What Oscillates? Displacement(y), Velocity(dy/dt), Acceleration(d2y/dt2)

Charge(q), current(dq/dt), dI/dt

Driving Agent Force Induced Voltage

Frequency

Simple Harmonic Motion is the projection of Uniform Circular Motion

Lissajous Figurecomponents in phase

Lissajous Figurecomponents out of phase

Lissajous Figurex 90o ahead of y

Lissajous Figurex 90o behind y