Simple harmonic motion

SIMPLE HARMONIC MOTION

B.TECH-CSE SECTION –A SEM. -

SUBMITTED BY:-

SACHIN JANGID

SUBMITTED TO:-Mr. UMESH KUMAR DWIVEDI

CONTENTS:

1. Periodic motion

2. Simple harmonic motion

3. Amplitude

4. Phase

5. Angular frequency

6. Period

7. Velocity of simple harmonic motion

8. Acceleration of simple harmonic motion

9. Energy in simple harmonic motion

10. Damped simple harmonic motion

11. Forced oscillations and resonance

Periodic motion

• Periodic (harmonic) motion – self-repeating motion

• Oscillation – periodic motion in certain direction

• Period (T) – a time duration of one oscillation

• Frequency (f) – the number of oscillations per unit time, SI unit of frequency 1/s = Hz (Hertz)

Tf

1

Heinrich Hertz(1857-1894)

Simple harmonic motion

Simple harmonic motion – motion that repeats itself and the displacement is a sinusoidal function of time

)cos()( tAtx

Amplitude

• Amplitude – the magnitude of the maximum displacement (in either direction)

)cos()( tAtx

Phase

)cos()( tAtx

Phase constant

)cos()( tAtx

Angular frequency

)cos()( tAtx0

)(coscos TtAtA )2cos(cos

)(cos)2cos( Ttt

T 2

T

2

f 2

Period

)cos()( tAtx

2

T

Differential equation of SHM

A differential equation is simply an equation containing a derivative. Since the motion is 1D, we can drop the vector arrows and use sign to indicate direction.

The constants k and m and both positive, so the k/m is always positive, always.For notational convenience, we write k/ m 2 . (The square on the reminds us that 2 is always positive.) The differential equation becomes

Fnet m a Fnet k x

2a dv / dt d2 x / dt2

and m a k x

d x -kx/m

This is the differential equation for SHM. We seek a solution y= y(t) to this equation, a function y = y(t) whose second time derivative is the function y(t) multiplied by a negative constant (2 = k/m). The way you solve differential equations is the same way you solve integrals: you guess the solution and then check that the solution works.

Based on observation, we guess a sinusoidal solution

x(t) A cost

where A, are any constants and (as we'll show) √k/m

d2 x / dt2 2 x

Velocity of simple harmonic motion

)cos()( tAtx

dt

tdxtv

)()(

dt

tAd )]cos([

)sin()( tAtv

Acceleration of simple harmonic motion

)cos()( tAtx

2

2 )()()(

dt

txd

dt

tdvta

)cos(2 tA

)()( 2 txta

The force law for simple harmonic motion

• From the Newton’s Second Law:

• •For simple harmonic motion, the force is proportional to the displacement

• Hooke’s law:

2mk

kxF

maF xm 2

m

k

k

mT 2

Energy in simple harmonic motion

• Potential energy of a spring:

• Kinetic energy of a mass:

2/)( 2kxtU )(cos)2/( 22 tkA

2/)( 2mvtK )(sin)2/( 222 tAm

)(sin)2/( 22 tkA km 2

Energy in simple harmonic motion

)()( tKtU

)(sin)2/()(cos)2/( 2222 tkAtkA

)(sin)(cos)2/( 222 ttkA

)2/( 2kA )2/( 2kAKUE

Pendulums

• Simple pendulum:

• Restoring torque:

• From the Newton’s Second Law:

• For small angles

)sin( gFL

I

sin

I

mgL

)sin( gFL

Pendulums


• On the other hand

L

at

I

mgL

L

s s

I

mgLa

)()( 2 txta

I

mgL

Pendulums


I

mgL 2mLI

2mL

mgL

L

g

g

LT

22

Simple harmonic motion and uniform circular motion

• Simple harmonic motion is the projection of uniform circular motion on the diameter of the circle in which the circular motion occurs

Pendulums

• Physical pendulum:

I

mgh

mgh

IT

22



)cos()( tAtx

dt

tdxtvx

)()(

)sin()( tAtvx



dt

tdxtvx

)()(

)sin()( tAtvx

)cos()( tAtx



2

2 )()(

dt

txdtax

)cos()( tAtx

)cos()( 2 tAtax

Where the force is proportional to the speed of the moving object and

acts in the direction opposite the motion.

The retarding force can be expressed as:

R = - bv ( where b is a constant called damping coefficient)

and the restoring force of the system is – kx,

then we can write Newton's second law as

xxx mabvkxF2

2

dt

xdm

dt

dxbkx

When the retarding force is small compared with the max restoring force that is, b is small the solution is,

)cos()( 2

tAetxt

m

b2)

2(

m

b

m

k

Damped simple harmonic motion

represent the position vs time for a damped oscillation with decreasing

amplitude with time

The fig. shows the position as a function in time of the object oscillation in

the presence of a retarding force, the amplitude decreases in time, this

system is know as a damped oscillator. The dashed line which defined the

envelope of the oscillator curve, represent the exponential factor

as the value of "b" increase the amplitude of the oscillations

decreases more and more rapidly.

When b reaches a critical value bc ( ), the system does

not oscillate and is said to be critically damped.

And when the system is overdamped.

oc mb 2/

oc mb 2/

The fig. represent position versus

time:

•under damped oscillator

•critical damped oscillator

- Overdamped oscillator.

For the forced oscillator is a damped oscillator driven by an external force that varies periodically Where

Forced oscillations

where ω is the angular frequency of the driving force and Fo is a constant

From the Newton's second law

tFtF o sin)(

2

2

sindt

xdmkx

dt

dxbtFmaF o

)cos( tAx

2222 )(

/

m

b

mFA

o

o

The last two equations show the driving force and the

amplitude of the oscillator which is constant for a

given driving force.

For small damping the amplitude is large when the

frequency of the driving force is near the natural

frequency of oscillation, or when ω ͌ ≈ ωo the is called

the resonance and the natural frequency is called the

resonance frequency.

is the natural frequency of the un-damped oscillator (b=0).m

ko

Amplitude versus the frequency, when the frequency of the

driving force equals the natural force of the oscillator,

resonance occurs. Note the depends of the curve as the value

of the damping coefficient b.

Simple harmonic motion

Engineering

Transcript of Simple harmonic motion