simple harmonic motion chapter 1 physics paper b bsc. i
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SIMPLE HARMONIC MOTION
Chapter 1
Physics Paper B BSc. I
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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.
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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.
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Example1
Mass-Spring System
a-is the accelerationa a a a
Equilibrium position
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Example2
aa
aa
Equilibrium position
Simple Pendulum
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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.
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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.
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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
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Displacement-Time Graph
y = rsin( wt)
t0
r
-r
y
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Velocity-Time Graphv = rwcos(wt)
t0
rw
- rw
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Acceleration-Time Graph
t0
a
rw2
-rw2
a = - rw2sin(t)
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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Compound Pendulum
Time Period is
where I is the moment of inertia of the pendulum.Centre of suspension and centre of oscillation are
interchangeable.
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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
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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
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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
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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
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Simple Harmonic Motion is the projection of Uniform Circular Motion
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Lissajous Figurecomponents in phase
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Lissajous Figurecomponents out of phase
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Lissajous Figurex 90o ahead of y
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Lissajous Figurex 90o behind y