➔ The defining equation of SHM
➔ Energy changes
Applications and skills
➔ Solving problems involving acceleration, velocity and displacement during simple harmonic motion, both graphically and algebraically
➔ Describing the interchange of kinetic and potential energy during simple harmonic motion
➔ Solving problems involving energy transfer during simple harmonic motion, both graphically and algebraically
➔ angular velocity–period equation: ω =2π/T
➔ defining equation for shm: a = -ω2x
➔ displacement–time equations: x = x0 sin ωt; x = x0 cos ωt
➔ velocity–time equations: v = ωx0 cos ωt; v = -ωx0 sin ωt
➔ velocity-displacement equation: v = ±ω √( x02 – x2 )
➔ kinetic energy equation: Ek =1/2 mω2( x02 – x2 )
➔ total energy equation: E = 1/2 mω2x02
➔ period of simple pendulum: T = 2π√ (l /g)
➔ period of mass–spring: T = 2π√ (m/k)
9.1 Simple Harmonic Motion (SHM)
Period of the motion is related to the angular frequency (ω), not to the amplitude or to the phase. T = 2π/ω.
Acceleration is proportional and in the opposite direction to the displacement. a = –ω²x.
Restoring force: After being moved a distance A (Amplitude) from equilibrium point, there will be a restoring force towards the center.
When there are two springs, the force doubles!
Period is independent of the amplitude/extension.
Small angle approximation: Acceleration is not proportional to the displacement (x). But if x is small (x < 10º) then sin(x/L) is approximately equal to x/L.
Period is independent of the mass.
When at t = 0, the displacement equals the amplitude:
x = xo cos(ωt);
v = -ωxo sin(ωt);
a = -ω²xo cos(ωt) = -ω²x.
When at t = 0, the displacement (x) equals zero:
x = xo sin(ωt);
v = -ωxo cos(ωt);
a = -ω²xo sin(ωt) = -ω²x.
Maximums: The maximum speed is ωxo and maximum acceleration is ω²xo.
Circular motion and SHM
Energy in SHM systems
Total energy conservation: ET = EK + EP.
ET = 1/2 mω²xo².
EK = 1/2 mv² = 1/2 mω² xo² sin²(ωt) = 1/2 mω² (xo² – x²).
EP = 1/2 mω² xo² cos²(ωt) = 1/2 mω²x².
PERIODIC AND OSCILLATORY MOTION
- Motion of moon around earth
- Motion of a piston in a cylinder
- Motion of a simple pendulum etc.
- The motion of a ball in bowl
- The needle of a sewing machine
- Vibrations of prongs of tuning fork etc.
- All oscillatory motion are periodic but all periodic motion are not oscillatory motion.
- The oscillatory motion which can be expressed in terms of sine and cosine function, is said to be harmonic motion.
SIMPLE HARMONIC MOTION (S.H.M.)
- Motion of a body suspended by a spring
- Oscillations of simple pendulum
EQUATIONS OF S.H.M.
TERMS RELATED TO S.H.M.
- Amplitude : The maximum displacement of the oscillating particle on either side of its mean position is called its amplitude. It is denoted by A.
- Time period : The time taken by a oscillating particle to complete one oscillation is called its time period. It is denoted by T.
- Frequency : It is the number of oscillations completed in one second.
It is denoted by υ.
- Angular frequency
- Phase : The parameter, by which the position of particle from its mean position is represented, is known as phase. The phase at any instant tells the state of position & direction of motion at that instant. The phase at time t = 0 is known as the initial phase or epoch (e).
- Total phase angle : The total angle (ωt + θ) is known as total phase angle.
CHARACTERISTICS OF S.H.M.
The displacement of a particle in S.H.M. is given by
The velocity of a particle in S.H.M. is given by
- Kinetic energy : A particle in S.H.M. possesses kinetic energy by virtue of its motion.
- Potential energy : A particle in S.H.M. possesses potential energy due to its displacement from the mean position.
- Total mechanical energy
- Restoring force F = – Mω2x
- Kinetic energy = (1/2) Mω2(A2 – x2)
- Potential energy = 1/2 Mω2x2
- Total energy of SHM = 1/2 Mω2A2
- The kinetic and potential energy of SHM varies sinusoidally with a frequency twice that of SHM.
- Total energy
- where ω is constant
- Geometrically the projection of the body undergoing uniform circular motion on the diameter of the circle is SHM.
- In a non-inertial frame.
SOME SYSTEMS EXECUTING S.H.M.
- When two springs having force constants k1 and k2 connected in parallel, then
- When two springs of force constants k1 and k2 are connected in series, then
- If two mass M1 and M2 are connected at the two ends of the spring, then their period of oscillation is given by
- When the length of spring increases, spring constant decreases. If the length of spring becomes n times, its spring constant becomes times and therefore time period will be increased by times.
- If we divide the spring into n equal parts, the spring constant of each part becomes n k. Hence time period when the same mass is suspended from each part is:
sin θ ≈ θ, so from equation (i)
where d = density of liquid, d′ = density of block, h = height of block
- In S.H.M. the phase relationship between displacement, velocity and acceleration, is as follows :
- The velocity is leading the displacement by a phase radian
- The acceleration is leading the displacement by a phase π radian
- The acceleration is leading the velocity by a phase radian.
- When , then velocity V = 0.86Vmax.
- When V = Vmax/2, the displacement x = 0.87A.
- When , the kinetic energy of S.H.M. is 75% of the total energy and potential energy 25% of the total energy.
- When the kinetic energy of S.H.M. is 50% of the total energy, the displacement is 71% of the amplitude.
- The time period of a simple pendulum of length l which is comparable with radius of earth.
- When l << R, then
- When l = R, we find
- When l = , then
- Under weightlessness or in the freely falling lift
This means, the pendulum does not oscillate at all.
- Time period of a simple pendulum in a train accelerating or retarding at the rate a is given by
- If a simple pendulum whose bob is of density do is made to oscillate in a liquid of density d, then its time period of vibration in liquid will increase and is given by
- The time period of a simple pendulum in a vehicle moving along a circular path of radius r and with constant velocity V is given by,
- If T1 and T2 are the time periods of a body oscillating under the restoring force F1 and F2 then the time period of the body under the influence of the resultant force will be
- (a) The percentage change in time period of simple pendulum when its length changes is
- If a wire of length l, area of cross-section A, Young’s modulus Y is stretched by suspending a mass m, then the mass can oscillate with time period
- If a simple pendulum is suspended from the roof of compartment of a train moving down an inclined plane of inclination θ, then the time period of oscillations
- If a ball of radius r oscillates in a bowl of radius R, then its time period is given by :
- If a disc of radius r oscillates about a point at its rim, then its time period is given by:
- The graph between the length of a simple pendulum and its time period is a parabola.
- The graph between the length of a simple pendulum and the square of its time period is a straight line.
- The graph between l & T and between l & T2 intersect at T = 1 second.
- The time period of the mass attached to spring does not change with the change in acceleration due to gravity.
- If the mass m attached to a spring oscillates in a non-viscous liquid density σ, then its time period is given by
- The length of second pendulum (T = 2 sec) is 99 cm
- The displacement, velocity and acceleration of S.H.M. vary simple harmonically with the same time period and frequency.
- The kinetic energy and potential energy vary periodically but not simple harmonically. The time period of kinetic energy or potential energy is half that of displacement, velocity and acceleration.
- The graph between displacement, velocity or acceleration and t is a sine curve. But the graph between P.E. or K.E. of S.H.M. and time t is parabola.
- If the bob of simple pendulum is -vely charged and a +vely charged plate is placed below it, then the effective acceleration on bob increases and consequently time period decreases.
- If the bob of a simple pendulum is -vely charged and is made to oscillate above the -vely charged plate, then the effective acceleration on bob decreases and the time period increases.
- A pendulum clock slows down in summer and goes faster in winter.
- Potential energy of a particle executing S.H.M. is equal to average force × displacement.
- If the total energy of a particle executing S.H.M. is E, then its potential energy at displacement x is
FREE, DAMPED, FORCED OSCILLATIONS AND RESONANCE
FORCED OSCILLATION AND RESONANCE