CLASS 11th
Oscillations
Oscillations
01. Periodic and Oscillatory Motions Periodic Motion Defined as that motion which repeats itself after equal intervals of time.
Oscillatory Motion Oscillatory or vibratory motion is defined as a periodic and bounded motion of a body about a fixed point.
Difference between Periodic and Oscillatory Motion Every oscillatory motion is periodic, but every periodic motion need not be oscillatory. e.g., circular motion (or the motion of planets around the sun) is a periodic motion, but it is not oscillatory because, the basic concept of to and fro motion about the mean position for oscillatory motion is not present here.
02. Periodic and Oscillatory Motions Period is the smallest interval of time after which the motion is repeated. Frequency is defined as the number of oscillations per unit time. It is the reciprocal of time period T. It is represented by the symbol v. The relation between v and T is ...(i)
03. Simple Harmonic Motion Consider a particle oscillating to and fro, about the origin of an x-axis, between the limits +A and –A as shown in figure.
This is considered simple harmonic motion if displacement x, of the particle from the origin, varies with time as (i) Where A, and are constants. SHM is not only periodic motion, but one in which displacement is a sinusoidal function of time.
04. Simple Harmonic Motion and Uniform Circular Motion Given figure describes the same situation mathematically. Let a particle P move uniformly in a circle of radius A with angular speed in anti-clockwise direction.
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Oscillations
The initial position vector of the particle makes an angle with positive direction of x-axis at t = 0. The position of P’ on the x-axis as the particle moves on the circle is given by Which is the defining equation of S.H.M.
Example
Find the time taken by the particle in going form x = 0 to x = where A is the amplitude.
sin
Solution
sin sin sin
⇒
05. Velocity and Acceleration in Simple Harmonic Motion The speed V of a particle in uniform circular motion is, its angular speed times the radius of the circle A. The direction of velocity at a time is along the tangent to the circle at the point where the particle is located at that instant. The instantaneous acceleration is than ...(i) Velocity in S.H.M. at instant is cos cos
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Oscillations sin
or ...(ii) Thus, the velocity of a particle in S.H.M. changes with displacement x of the particle, when the displacement is zero (x = 0), i.e., when it passes through its equilibrium position, velocity is maximum max and when the displacement is maximum (y = A), velocity is zero (v = 0). sin ∝ Thus, acceleration of SHM is proportional to the displacement x and its direction is opposite to the direction of displacement from the equilibrium. The period is defined as the time taken by the particle executing SHM to complete one vibration.
06. Energy in Simple Harmonic Motion KE and PE of a particle in SHM vary between zero and their maximum values. The velocity of a particle executing SHM is zero at the extreme positions (it is a periodic function of time). So, the kinetic energy (K) of such a particle is sin sin ...(i) This is also a periodic function of time, being zero when the displacement is maximum and minimum when the particle is at the mean position. The concept of potential energy is possible only for conservative forces. The spring force, F = -kx is a conservative force, with associated potential energy. ...(ii) So, the P.E. of a particle executing SHM is, cos ...(iii)
So, PE of a particle executing SHM is also periodic, with period , being zero at the mean position and maximum at the extreme displacements.
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Oscillations From equations (i) & (iii), we get The total energy, E, of the system is E = U + K cos sin cos sin ...(iv) The total mechanical energy of a harmonic oscillator is thus independent of time as expected for motion under any conservative force. The time and displacement dependence of PE and KE of a linear simple harmonic oscillator are shown in figure.
(b)
(a)
K.E., P.E. and total energy as a function of time (shown in (a) and displacement (shown in (b)) of a particle
in SHM. The K.E. and P.E. both repeat after a period The total energy remains constant at all t or x. Both KE and PE in SHM are seen to be always positive in figures. K.E. can never be negative as it is proportional to the square of speed. P.E. is positive by choice of the undermined constant in P.E. is positive by choice of the undermined constant in P.E. Both KE and PE reach maximum value twice, during each period of SHM. For x = 0, the energy is kinetic, at the extremes ± , it is all potential energy. In the course of motion between these limits, KE increases at the expense of PE or vice-versa.
07. The Simple Pendulum A small bob of mass m is tied to an inextensible massless string of length L. The other end of the string is fixed to a support on the ceiling. The bob oscillates in a plane about the vertical line through the support.
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Oscillations
(a) (b) ...(i) This is a restoring torque that tends to reduce angular displacement, hence, the negative sign. ...(ii) ...(iii) or ...(iv) When is small, sin can be approximated by ...(v) Calculate the length of a Second’s pendulum (the pendulum which ticks seconds)
Example Solution
× = × = 1 m (As time period of a seconds pendulum (i.e., the pendulum which ticks seconds) is 2s.
08. Effect of change in length so following discussion on it The period of the simple pendulum is proportion to is important. so the graph between T and will be a parabola while between and (a) Since ∝ will be a straight line. (b) If the bob is hollow sphere full of water and water comes out slowly through a hole at the bottom, the time period will first increase (less number of oscillation per second will be made) then the time period will be restored to original value once the total water has come out, (It is due to change of position of centre of mass of bob due to leakage of water).
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Oscillations ∆ ∆ ∴ Now if the temperature of a simple pendulum changes by ∆ and its thermal coefficient ∆ ∆ ∆ of linear expansion is then ∆ (Note ∆ is small). So
∝
(c) Since
Effect of buoyancy If the solid bob of the pendulum has relative density D and has been submerged in a non-viscous liquid of relative density then effective acceleration due to gravity g’ where So, the time period will increase.
Example
If the length of a simple pendulum of a clock increases by 2%, how much loss or gain of time (in second) per day will take place?
Solution
% change is very less.
∆
∆
So, Here, T is time in second for the whole day = 86400 s
∆
Given
∴ ∴
∆ × ∆
As length is increasing, the time period T will become large and the clock will go slow, so it will lose 864 s.
09. Damped Simple Harmonic Motion ...(i) The solution of equation describes the motion of the block under the influence of damping force which is proportional to velocity. The solution is of the form
cos′ ...(ii) where a is the amplitude and ′ is the angular frequency of the damped oscillator given by ′ ...(iii) The solution, can be graphically represented as shown in given figure.
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Oscillations
10. Forced Oscillations and Resonance Let an external force F(t) of amplitude F0 that varies periodically with time be applied to a damped oscillator. It can be represented as cos ...(i) The motion of a particle under the combined action of a linear restoring force, damping force and a time dependent driving force represented by equation (i) is given by, cos ...(ii) Substituting for acceleration in equation and rearranging it, we get cos ...(iii) ...(iv) where t is the time measured from the moment, the periodic force is applied. The amplitude A is a function of the forced frequency and the natural frequency . It is given by
tan
and
...(v)
...(vi)
Case-I Small damping, driving frequency far from natural frequency is much smaller than and it can be neglected.
...(vii)
Case-II Driving frequency close to natural frequency If is very close to is much less than for any reasonable value of band, hence Equation reduces to
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...(viii)
Oscillations The maximum possible amplitude for a given driving frequency is controlled by the driving frequency and the damping, and is never infinity. The phenomenon of increase in amplitude when the driving force is close to the natural frequency of the oscillator is called resonance.
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Oscillations
AIIMS Exercise (1)
1. A particle of mass m is subjected to two SHM’s given by y1 = 4 sin and y2 = 2 sin . Find maximum speed of the particle.
(a) ω (b) 2 ω
(c) ω/2 (d) 4 ω
2. When displacement of a particle executing SHM is half of the amplitude, its potential energy is 5 J. Find the total energy of the particle in SHM. (a) 20 J (b) 10 J
(c) 30 J (d) 25 J
3. The length of a pendulum decreases by 36%. Find the percentage decrease in its time period. (a) 18% (b) 20%
(c) 22% (d) 36%
4. If the length of a pendulum decreases by 2%. Find the percentage decrease in its time period. (a) 2% (b) %
(c) 1% (d) 4%
5. If two simple harmonic motions are represented by equations y1 = 10 sin and .cos 3πt], find the ratio of their amplitudes. y2 = 5 [sin 3πt +
(a)
(c)
(b)
(d)
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Oscillations 6. Two particles execute SHM of the same amplitude and frequency along the same straight line. They pass one another when going in opposite directions, each time their displacement is half of their amplitude. Find the phase difference between them. (a) rad (b) rad
(c) rad (d) rad
7. The maximum velocity of a body executing SHM is v0. Find the average velocity of the particle from one extreme position to other extreme position. (a) (b)
(c) (d)
8. Two simple pendulums of length 1 m and 16 m respectively are both given small displacements in the same direction at the same instant. After how many oscillations of the shorter pendulum, they will again be in phase? (a) (b)
(c) 3 (d) 4
9. The mass and diameter of a planet are twice those of earth. The period of oscillation of pendulum on this planet will be (if it is a second’s pendulum on earth) : second (a) (b) second
(c) 2 second (d) (1/2) second
10. The time period of a particle in simple harmonic motion is 8 seconds. At t = 0 it is at the mean position. The ratio of the distances travelled by it in the first and second seconds is : (a) (b)
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(c) (d)
Oscillations In each of the following questions, a statement of Assertion (A) is followed by a corresponding statement of Reason (R). Of the following statements, choose the correct one. (a) Both A and R are true and R is the correct explanation of A. (b) Both A and R are true but R is not correct explanation of A. (c) A is true but R is false. (d) A is false but R is true. (e) Both A and R are false.
11. (A) : In SHM the maximum kinetic energy of the particle = Maximum potential energy of the particle. (R) : the equation of motion of a particle executing SHM is y = A sin ωt. 12. (A) : The pendulum clock runs slow on the surface of moon than on the surface of earth. (R) : There is no atmosphere on the surface of moon. 13. (A) : If a man has a wrist watch in his hand and he jumps from a high tower, then during fall, his wrist watch will keep correct time. (R) : In spring system and T is independent of the value of g.
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