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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