simple harmonic motion
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SIMPLE HARMONIC MOTIONDr. Popat S. TambadeAssociate ProfessorProf. Ramkrishna More Arts, Commerce and Science CollegeAkurdi, Pune 411 044
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Content EquilibriumStable equilibriumUnstable EquilibriumOscillatory MotionSpring Mass systemSimple harmonic MotionDisplacement and velocityPeriodic Time FrequencyDisplacement and AccelerationEnergy of SHMLissajous FiguresAngular SHMSimple Pendulum
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EquilibriumTypes of equilibriumsStable EquilibriumUnstable equilibriumNeutral equilibriumThe body is said to be in equilibrium at a point when net force acting on the body at that point is zero.C
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Stable equilibrium
If a slight displacement of particle from its equilibrium position results only in small bounded motion about the point of equilibrium, then it is said to be in stable equilibrium
Equilibrium positionC
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Potential energy curve for stable equilibrium
-a+a
- x
x
0
V(x)x
Slope =dVdx
Tangent at AAPositive F = dVdx
ForceFForce is negative i.e. directed towards equilibrium positionB
Tangent at B
Slope =dVdx
NegativeForce is positive i.e. directed towards equilibrium positionFSimulationC
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Unstable equilibrium
If a slight displacement of the particle from its equilibrium position results unbounded motion away from the equilibrium position, then it is said to be in unstable equilibrium
Equilibrium positionC
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Potential energy curve for unstable equilibrium
-a+a
- x
x
0
V(x)x
Slope =dVdx
Tangent at AANegative F = dVdx
ForceFForce is positive i.e. directed away from equilibrium positionB
Tangent at B
Slope =dVdx
PositiveForce is negative i.e. directed away from equilibrium positionFC
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Oscillatory MotionAny motion that repeats itself after equal intervals of time is called periodic motion.
If an object in periodic motion moves back and forth over the same path, the motion is called oscillatory or vibratory motion
C
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Spring-Mass system
m
m
x = 0
x
x Relaxed modeExtended modeCompressed mode
F
FmWe know that for an ideal spring, the force is related to the displacement by
C
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Simple Harmonic Motion
Linear simple harmonic motion : When the force acting on the particle is directly proportional to the displacement and opposite in direction, the motion is said to be linear simple harmonic motionDifferential equation of motion ismd2xdt2
+ kx = 0
d2xdt2
+ 2 x = 0
whereSolution is x = a sin (t + )(t + ) is called phase and is called epoch of SHM
C
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a and are determined uniquely by the position and velocity of the particle at t = 0If at t = 0 the particle is at x = 0, then = 0If at t = 0 the particle is at x = a, then = /2The phase of the motion is the quantity (t + )x (t) is periodic and its value is the same each time t increases by 2 radiansx = a sin (t + )The displacement of particle from equilibrium position is C
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Simple harmonic motion (or SHM) is the sinusoidal motion executed by a particle of mass m subject to one-dimensional net force that is proportional to the displacement of the particle from equilibrium but opposite in sign C
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x = a sin (t + )Equation of SHM isThe velocity is v = dxdt
v = a cos (t + )orv =
The velocity is zero at extreme positions and maximum at equilibrium positionC
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Graphs of Displacement and Velocity
xt, time
TTt
232
52
2
372
4
2
2
v
For = 2
x = a sin (t + )v = a cos (t + )The phase difference between velocity and displacement is 2
+a-a
T TT = 2, The period of oscillation is T = 2/ T is called periodic timeC
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Periodic Time
The period of SHM is defined as the time taken by the oscillator to perform one complete oscillationAfter every time T, the particle will have the same position, velocity and the direction
TmC
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The frequency represents the number of oscillations that the particle undergoes per unit time intervalThe inverse of the period is called the frequency
Units are cycles per second = hertz (Hz)
Frequency C
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The frequency and the period depend only on the mass of the particle and the force constant of the spring They do not depend on the parameters of motion like amplitude of oscillationThe frequency is larger for a stiffer spring (large values of k) and decreases with increasing mass of the particle
C
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Displacement and acceleration
xt
232
52
2
372
4
Ax
For = 2
x = a sin (t + )A = - a2 sin (t + )The phase difference between acceleration and displacement is
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EnergyThe potential energy is V = k x212
The kinetic energy is K = m v 212
orK = m 2 (a2 x2)12
The total energy is E = K + VorE = m 2 a2 12
Thus, total energy of the oscillator is constant and proportional to the square of amplitude of oscillationsC
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x -a 0 +a
Amaxa
Summary .Caavmaxvmax
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-a+ax
0P. E. K. E.P. E. =K. E.
a/ 2
- a/ 2
EnergyGraphical Representation of K. E. and P. E.E = m 2 a2 12
The total mechanical energy is constant The total mechanical energy is proportional to the square of the amplitudeEnergy is continuously being transferred between potential energy stored in the spring and the kinetic energy of the blockC
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Variation of K.E. and P. E. With time
xt
232
52
2
372
4
For = 2
x = a sin (t + )
t
0E
V = k x212
K = m 2 (a2 x2)12
For one cycle of oscillation of particle there are two cycles for K. E. and P.E.. Thus frequency of K. E. or P. E. is 2nP. E. K. E.
0C
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Angular SHMIf path of particle of a body performing an oscillatory motion is curved, the motion is known as angular simple harmonic motionDefinition : Angular simple harmonic motion is defined as the oscillatory motion of a body in which the body is acted upon by a restoring torque (couple) which is directly proportional to its angular displacement from the equilibrium position and directed opposite to the angular displacement
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k is the torsion constant of the support wireThe restoring torque isNewtons Second Law gives Angular SHM
I moment of inertia
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The torque equation produces a motion equation for simple harmonic motionThe angular frequency is
The period is
No small-angle restriction is necessaryAssumes the elastic limit of the wire is not exceeded
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Simple PendulumThe equation of motion is
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When angle is very small, we have sin q q
The Period is
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When q > 100 but q < 200 , then period is When q > 200 , then period is
But when angular arc is not small ,then we have to solve
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Dr. P. S. Tambade received an outstanding paper award in E-Learn 2008, Las Vegas
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Thank You
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txvK. E.P. E.E
0+ a00 eq \f(1,2) m 2a2 eq \f(1,2) m 2a2
T/40 a eq \f(1,2) m 2a2 0 eq \f(1,2) m 2a2
T/2 a00 eq \f(1,2) m 2a2 eq \f(1,2) m 2a2
3T/40+ a eq \f(1,2) m 2a2 0 eq \f(1,2) m 2a2
T+a00 eq \f(1,2) m 2a2 eq \f(1,2) m 2a2