SIMPLE HARMONIC MOTION - PowerPoint Presentation

Slide1

SIMPLE HARMONIC MOTION

Chapter 1

Physics Paper B

BSc

. ISlide2

Motion of a body

PERIODIC MOTION- The motion which repeats itself at a regular intervals of time is known as Periodic Motion.

Examples are:Revolution of earth around sun

The rotation of earth about its polar axis

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:

Motion of bob of a simple pendulum

Motion of a loaded spring

Motion of the liquid contained in U-tube

All oscillatory motions are periodic but all periodic motions are not oscillatory.Slide3

Simple Harmonic Motion (S.H.M)

DEFINITION S.H.M is a motion in which restoring force is

directly proportional to the displacement of the particle from the mean or equilibrium position . 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.Slide4

Example1

Mass-Spring Systema-is the acceleration

a

a

a

a

Equilibrium

positionSlide5

Example2

a

a

a

a

Equilibrium

position

Simple PendulumSlide6

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.Slide7

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.Slide8

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:

TSlide9

Displacement-Time Graph

y =

rsin

(

w

t)

t

0

r

-r

ySlide10

Velocity-Time Graph

v =

r

w

cos

(

w

t

)

t

0

r

w

-

r

wSlide11

Acceleration-Time Graph

t

0

a

r

w

2

-

r

w

2

a =

-

r

w

2

sin(

t

)Slide12

Phase Difference

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

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

o

.

Fig.3 shows two waves having phase difference of

/4 or 45

o

.Slide13

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.Slide14

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.Slide15

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.Slide16

Simple Pendulum

mgsin

q

mgcos

q

q

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.Slide17

Simple Pendulum

Where T is time period of pendulum.Slide18

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.Slide19

Compound Pendulum

Time Period iswhere I is the moment of inertia of the pendulum.

Centre of suspension and centre of oscillation are interchangeable.Slide20

Torsional Pendulum

If 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

Slide21

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 OscillatorSlide22

Energy of Electrical OscillatorIn an electrical oscillator we have two types of energies:

Electrical energy stored in capacitorMagnetic energy stored in inductorTotal energy of electrical oscillator at any instant of time is Slide23

Comparison of Mechanical and Electrical Oscillator

Parameter

Mechanical OscillatorElectrical 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)InertiaMass

mInductance LElasticityStiffness k1/CWhat Oscillates?Displacement(y), Velocity(

dy/dt), Acceleration(d2y/dt2)Charge(q), current(dq/dt

), dI/dtDriving AgentForceInduced VoltageFrequencySlide24

Simple Harmonic Motion is the projection

of Uniform Circular MotionSlide25

Lissajous Figurecomponents

in phaseSlide26

Lissajous Figurecomponents

out of phaseSlide27

Lissajous Figurex 90

o ahead of ySlide28

Lissajous Figurex 90

o behind y