Objective questions of wave

1. The amplitude of a wave disturbance propagating in the positive direction is given by y=1/(1+x2) at time t=0 and by y=1/[1+(x-5)2] at t=5 seconds where x and y are in meters.The shape of the wave disturbance does not change during the propagation.The velocity of the wave is
a. 1 m/sec
b. 1.5 m/sec
c. .5 m/sec
d. 2 m/s


2.A transverse wave in a medium is described by the equation

y=Asin2(wt-kx).
The magnitude of the maximum velocity of particles in the medium is equal to that of the wave velocity.if the value of A is
a.λ/2π
b λ/4π
c. λ/π
d 2λ/π


3.A plane progressive wave is represented by the equation
y=cos(2πt-πx)

The equation of the wave with triple of the amplitude and double the frequency
a. y=3cos(4πt-πx)
b. y=3cos(5πt-πx)
c. y=3cos(4πt+πx)
d. y=3cos(3πt-πx)


4. In the above example ,The equation of wave with double of the amplitude and double the frequency but travelling in the opposite direction
a. y=2cos(4πt+πx)
b. y=2cos(5πt-πx)
c. y=2cos(4πt-πx)
d. y=2cos(3πt-πx)

5.The displacement of a particle having wave motion given by
y=cos2(t/4)sin(50t)
This expression may be considered to be a result of the superposition of how many wave motions
a. one
b Two
c. Three
d. Five


6.A wave is represented by the equation
y=(1mm)sin[(60 s-1)t+(4 m-1)x]

which one of the following is true
a. Frequency =30/π
b Amplitude=.001mm
c. Maximum Velocity of the Particle 60 mm/sec
d. wave velocity is 100m/s


7.the displacement of the particles in a string streched in the x-direction is represented by y.Among the following expressions for y,those describing wave motion ares
a. cospxsinqt
b p2x2-w2t2
c.cos2(px+wt)
d. cos(p2x2-w2t2)



8.A transverse wave on a string,the string displacement is described as
y(x,t)=1/1+(x-at)2

where a is negative constant
which of the following is true
a. The Shape of the string at t=0 is y=/1+x2
b. The shape of the waveform does not change as its move along the string
c Waveform moves in the -x direction
d. The speed of the waveform is |a|

Solutions

Please take a look at these related posts also
SHM concept Part 1
SHM concept Part 2
Conceptaul Question for SHM
Subjective questions for SHM
objective Question for SHM
Waves Concept part 1
Waves Concept part 2
Waves Concept part 3
Conceptaul Question for waves
Subjective Question for waves

Physics Syllabus for IITJEE

General:

Units and dimensions
Dimensional analysis
least count
significant figures
Methods of measurement and error analysis for physical quantities pertaining to the following experiments: Experiments based on using Vernier calipers and screw gauge (micrometer), Determination of g using simple pendulum, Young’s modulus by Searle’s
method, Specific heat of a liquid using calorimeter, focal length of a concave mirror and a convex lens using u-v method, Speed of sound using resonance column, Verification of Ohm’s law using voltmeter and ammeter, and specific resistance of the material of a wire using meter bridge and post office box.

Mechanics I:

Kinematics in one and two dimensions (Cartesian coordinates only), projectiles
Uniform Circular motion
Relative velocity.
Newton’s laws of motion
Inertial and uniformly accelerated frames of reference
Static and dynamic friction
Kinetic and potential energy
Work and power
Conservation of linear momentum and mechanical energy.
Systems of particles
Centre of mass and its motion
Impulse
Elastic and inelastic collisions.
Law of gravitation
Gravitational potential and field
Acceleration due to gravity
Motion of planets and satellites in circular orbits
Escape velocity.

Mechanics II:

Rigid body
moment of inertia
parallel and perpendicular axes theorems
moment of inertia of uniform bodies with simple geometrical shapes
Angular momentum
Torque
Conservation of angular momentum
Dynamics of rigid bodies with fixed axis of rotation
Rolling without slipping of rings
cylinders and spheres
Equilibrium of rigid bodies
Collision of point masses with rigid bodies.
Hooke’s law
Young’s modulus.

FLUID MECHANICS:

Pressure in a fluid
Pascal’s law
Buoyancy
Surface energy and surface tension
capillary rise
Viscosity (Poiseuille’s equation excluded)
Stoke’s law
Terminal velocity
Streamline flow
equation of continuity
Bernoulli’s theorem and its applications.

WAVES AND OSCILLATION:

Linear and angular simple harmonic motions.
Wave motion (plane waves only)
longitudinal and transverse waves
superposition of waves
Progressive and stationary waves
Vibration of strings and air columns
Resonance; Beats
Speed of sound in gases
Doppler effect (in sound).

Thermodynamics:

Thermal expansion of solids, liquids and gases
Calorimetry
latent heat
Heat conduction in one dimension
Elementary concepts of convection and radiation
Newton’s law of cooling
Ideal gas laws
Specific heats (Cv and Cp for monoatomic and diatomic gases)
Isothermal and adiabatic processes
bulk modulus of gases
Equivalence of heat and work
First law of thermodynamics and its applications (only for ideal gases)
Blackbody radiation
absorptive and emissive powers
Kirchhoff’s law
Wien’s displacement law
Stefan’s law.

Electricity And Magnetism:

Coulomb’s law
Electric field and potential
Electrical potential energy of a system of point charges and of electrical dipoles in a uniform electrostatic field
Electric field lines
Flux of electric field
Gauss’s law and its application in simple cases, such as, to find field due to infinitely long straight wire
uniformly charged infinite plane sheet and uniformly charged thin spherical shell.
Capacitance
Parallel plate capacitor with and without dielectrics
Capacitors in series and parallel
Energy stored in a capacitor.
Electric current
Ohm’s law; Series and parallel arrangements of resistances and cells; Kirchhoff’s laws and simple applications; Heating effect of current.
Biot–Savart’s law and Ampere’s law
Magnetic field near a current-carrying straight wire, along the axis of a circular coil and inside a long straight solenoid;
Force on a moving charge and on a current-carrying wire in a uniform magnetic field.
Magnetic moment of a current loop
Effect of a uniform magnetic field on a current loop
Moving coil galvanometer, voltmeter, ammeter and their conversions.
Electromagnetic induction
Faraday’s law, Lenz’s law
Self and mutual inductance
RC, LR and LC circuits with d.c. and a.c. sources.

Optics:

Rectilinear propagation of light
Reflection and refraction at plane and spherical surfaces
Total internal reflection
Deviation and dispersion of light by a prism
Thin lenses
Combinations of mirrors and thin lenses
Magnification.
Wave nature of light
Huygen’s principle, interference limited to Young’s double-slit experiment.

Modern physics:

Atomic nucleus
Alpha, beta and gamma radiations
Law of radioactive decay
Decay constant
Half-life and mean life
Binding energy and its calculation
Fission and fusion processes
Energy calculation in these processes.
Photoelectric effect
Bohr’s theory of hydrogen-like atoms
Characteristic and continuous X-rays
Moseley’s law
de Broglie wavelength of matter waves.

Please take a looks at these post also
Books for IITJEE Physics
Books for AIEEE Physics
Study Tips for IITJEE PART 1

Conceptual Questions of waves

1. Transverse wave velocity in a stretched string depends on
a. frequency of wave
b. tension
c. length of string
d. linear mass density string


2. A transverse wave travels along the x axis.The particles of the medium must move
a. Along the z-axis
b Along the x-axis
c. In the Y-Z plane
d. Along the y axis



3.What is true for a standing wave on the string
a.All the particles are never at rest simultaneously
b. In one complete cycle,all the particles cross their mean position simultaneously twice
c. In one complete cycle,all the particles cross their mean position simultaneously once
d. All the particles acquire their positive extreme positions simultaneously once in a cycle



4.Choose the incorrect one
a. When a ultrasonic wave travels from air into water.It bends towards the normal to the air-water interface
b.Any function of the form y(x,t)=f(vt+x) represents a travelling wave
c.the velocity ,wavelenght and frequency of wave undergo change when it is reflected from a surface
d. None of the above



5.Match the following with the types of the wave

A) Thermal radiation received from the sun
B)Sound waves produced by the vibrating string of guitar
C) Radio waves sent out from broadcasting station
D)X Rays
E) Waves produced in the air by the vibrating tuning fork

P) longitudinal
Q) Transverse





6. Which of the following functions represent a travelling wave
a. y=pcos(qx)sin(rt)
b. y=psin(qx+rt)
c. y=psin(qx-rt)
d. none of the above


7. Which of the following is not a standing wave
a. y=pcos(qx)sin(rt)
b. y=psin(qx+rt)+psin(qx-rt)
c. y=psin(qx+rt)
d Non of the above



8. When a wave is refracted into another medium which of the following will change
a. Velocity
b. Frequency
c. Phase
d. Amplitude

9.A pipe closed at one end and open at other will give
a. All even harmonics
b, All odd harmonics
c. All the harmonics
d. None of the harmonics



10. To raise the pitch of a stringed musical instrument ,the player can
a. Lossen the string
b. Tighten the string
c. Shorten the string
d. Lengthen the string

Solutions

Waves Concept

PART 3

Beats:-
-Interference of two harmonic waves of different frequencies and wavelength produces beats.
-In beats phenomenon interfering harmonic waves have slightely differing frequencies ν1 and ν2 such that
12|<<(ν12)/2
-Thus beats arises when two waves having slightly differing frequencies ν1 and ν2 and comparable amplitude are superposed. The beat frequency is
νbeat= ν1 ∼ ν2
-Musicians use beat phenomenon for tuning their instruments.
-For tuning an instrument for certain standard frequency it is sounded against a standard frequency and it is tuned untill the beats get disappeared.

Doppler effect:-
-Doppler effect is a change in the observed frequency of the wave when the source s and the observer o moves relative to the medium.
-There are three different ways where we can analyse this change in frequency.
(1) When observer is stationary and source is moving then change in frequency for source aproaching observer is
ν=ν0(1+vs/v)
where, vs=velocity of source relative to the medium
v=velocity of wave relative to the medium
ν=observed frequency of sound waves in term of source frequency
ν0=source frequency
-Change in frequencywhen source receeds from stationary observer is
ν=ν0(1-vs/v)
-Observer at rest measures higher frequency when source aproaches it and it measures lower frequency when source receeds from the observer.
(2)Doppler effect in frequency when observer is moving with a velocity vo towards source and the source is at rest is
ν=ν0(1+vo/v)
(3) If both source and observer are moving then frequency observed by observer is
ν=ν0(v+vo)/(v+vs)
and all the symbols have respective meanings as told earlier.

Galileo Biograpgy

Galileo Galilei (1564-1642)


Galileo's experiments into gravity refuted Aristotle Galileo was a hugely influential Italian astronomer, physicist and philosopher.

Galileo Galilei was born on 15 February 1564 near Pisa, the son of a musician. He began to study medicine at the University of Pisa but changed to philosophy and mathematics. In 1589, he became professor of mathematics at Pisa. In 1592, he moved to become mathematics professor at the University of Padua, a position he held until 1610. During this time he worked on a variety of experiments, including the speed at which different objects fall, mechanics and pendulums.

In 1609, Galileo heard about the invention of the telescope in Holland. Without having seen an example, he constructed a superior version and made many astronomical discoveries. These included mountains and valleys on the surface of the moon, sunspots, the four largest moons of the planet Jupiter and the phases of the planet Venus. His work on astronomy made him famous and he was appointed court mathematician in Florence.

In 1614, Galileo was accused of heresy for his support of the Copernican theory that the sun was at the centre of the solar system. This was revolutionary at a time when most people believed the Earth was in this central position. In 1616, he was forbidden by the church from teaching or advocating these theories.

In 1632, he was again condemned for heresy after his book 'Dialogue Concerning the Two Chief World Systems' was published. This set out the arguments for and against the Copernican theory in the form of a discussion between two men. Galileo was summoned to appear before the Inquisition in Rome. He was convicted and sentenced to life imprisonment, later reduced to permanent house arrest at his villa in Arcetri, south of Florence. He was also forced to publicly withdraw his support for Copernican theory.

Although he was now going blind he continued to write. In 1638, his 'Discourses Concerning Two New Sciences' was published with Galileo's ideas on the laws of motion and the principles of mechanics. Galileo died in Arcetri on 8 January 1642.

Waves and Oscillation Index

Waves and Oscillation

SHM concept Part 1
SHM concept Part 2
Conceptaul Question for SHM
Subjective questions for SHM
objective Question for SHM
Waves Concept part 1
Waves Concept part 2
Waves Concept part 3
Conceptaul Question for waves
Objective Question for waves
Subjective Question for waves
IITJEE Test Series I

Mechanics Index

Mechanics Index

Gravitation Concept
Conceptaul Question for Gravitation
Motion in a Plane
Conceptaul Question for Motion in a plane
objective Question for kinematics
Subjective Question for kinematics
Relative Velocity Concept
Conceptaul Question for relative velocty
Graphical Question for Motion in a Plane
Newton's law of Motion Concept
Conceptaul Question for Newton law of motion
Uniform Cicular Motion-Friction-Frame of Refrence
Work And Energy
Momentum and Center of Mass
IIT JEE test series
Problem Solving tips for Mechanics
Challenging Problems Mechanics
15 min Test series
30 min Test series
PMT test series
Rotational Motion -I
Rotational Motion -II
Rotational Objective


Note: solution are available at the bottom of posts

Thermal Index

Thermodynamics Index.

Thermal Expansion
Study Tips Part 1
Study Tips Part 2
Thermo Quick Recap
Conceptual Questions Part 1
Conceptual Questions Part 2
Conceptual Questions Part 3
Conceptual Questions Part 4
IITJEE Objective Questions part 1
IITJEE Objective Questions Part 2
Thermal Expansion Problems
IITJEE Subjective Questions
Thermo Questions
Past Year AIEEE Thermo Questions
IITJEE test series 1
IITJEE test series 2



Note:Solutions are available at the bottom of each post

Waves Concept

PART2

Interference of waves:-
-From principle of superposition we know that overlaping waves algbrically add togather to produce a net wave without altering the way of each other or the individual waves.
-If two sinusoidal waves of the same amplitude and wavelength travell in the same direction they interfere to produce a resultant sinusoidal wave travelling in that direction.
-The resultant wave due to interference of two sinusoidal waves is given by the relation
y′(x,t)=[2Amcos(υ/2)]sin(ωt-kx+υ/2)where υ is the phase difference between two waves.
-If υ=0n then there would be no phase difference between the travelling waves and the interference would be fully constructive.
-If υ=π then waves would be out of phase and there interference would be distructive.

Reflection of waves:-
-When a apulse or travelling wave encounters any boundary it gets reflected.
-If the boundary is not completely rigid then then a part of wave gets reflected and rest of it's part gets transmitted or refracted.
-A travelling wave at a rigid boundary is reflected with a phase reversal but the reflection at open boundary takes place without phase change.
-if an incident wave is represented by
yi(x,t)=A sin(ωt-kx)then reflected wave at rigid boundary is
yr(x,t)=A sin(ωt+kx+π)
=-Asin(ωt+kx)
and for reflections at open boundary reflected wave is given by
yr(x,t)=Asin(ωt+kx)
Standing waves:-
-The interference of two identical waves moving in opposite directions produces standing waves.
-For a string with fixed ends standing wave is given by
y(x,t)=[2Acos(kx)]sin(ωt)above equation does not represent travelling wave since it does not have characterstic form involving (ωt-kx) or (ωt+kx) in the argument of trignometric function.
-In standing waves amplitude of waves is different at different points i.e., at nodes amplitude is zero and at antinodes amplitude is maximumwhich is equal to sum of amplitudes of constituting waves.
-At intermediate points amplitude of wave varies between these two limits of maxima and minima

Normal modes of stretched string:--Frequency of transverse motion of stretched string of length L fixed at both the ends is given by
f=nv/2L
where n=1,2,3,4,.......
-The set of frequencies given by above relation are called normal modes of oscillation of the system.
-The mode with n=1 is called the fundamental mode with frequancy
f1=v/2L-Similarly second harmonic is the oscillation mode with n=2 and so on.
-Thus the string has infinite number of possible frequency of viberation which are harmonics of fundamental frequency f1 such that fn=nf1

Waves concept

-Definition of wave:-
It is a disturbance which travels through the medium due to repeated periodic motion of particles of the medium about their equilibrium position.

-Example of wave motion are sound waves traveling through an intervening mediun, water waves, light waves and many more such examples are there.

-Waves requiring material medium for their propagation are called MECHANICAL WAVES. Mechanical waves are governed by Newton's law of motion.

-Sound waves are mechanical waves in atmosphere between source and the listner and require medium for their propagation.

-Other examples of mechanical waves are sesmic waves and water waves.

-Those waves which does not require material medium for their propagation are called NON MECHANICAL WAVES.

-One familiar example of NON MECHANICAL WAVES is waves associated with light or light waves. Another such examples are radio waves, X-rays, micro waves, UV light, visible light and many more.

-Transverse waves are such waves where the displacements or oscillations are perpandicular to the direction of propagation of wave.

-Longitudinal waves are those waves in which displacement or oscillations in medium are parallel to the direction of propagation of wave for example sound waves.

-At any time t , displacement y of the particle from it's equilibrium position as a function of the coordinate x of the particle is
y(x,t)=A sin(ωt-kx)
where,
A is the amplitude of the wave
k is the wave number
ω is angular frequency of the wave
and (ωt-kx) is the phase.

-Wavelength λ and wave number k are related by the relation
k=2π/λ
-Time period T and frequency f of the wave are related to ω by
ω/2π = f = 1/T
-speed of the wave is given by
v = ω/k = λ/T = λf

-Speed of a transverse wave on a stretched string depends on tension and the linear mass density of the string not on frequency of the wave
i.e,
v=√T/μ
T=Tension in the string
μ=Linear mass density of the string

-Sounds waves are longitudinal mechanical waves that can travel through solids,liquid and gases

-Speed of longitudinal waves in a medium is given by
v=√B/ρ
B=bulk modulus
ρ=Density of the medium

-Speed of longitudinal waves in ideal gas is
v=√γP/ρ

P=Pressure of the gas
ρ=Density of the gas
γ=Cp/CV

Principle of superposition:
When two or more waves traverse thrugh the same medium,the displacement of any particle of the medium is the sum of the displacement that the individual waves would give it.

y=Σyi(x,t)

Solutions for Conceptual questions of Newton's law

1.sol:- (d) A body acted uopn by a certain force produces acceleration i.e. it undergoes change in it's velocity. hence choice (d) is correct
2. sol:- (a)
3. sol:-(d)
4. sol:- (b)
since velocity of block when it reaches the ground is given by v=(2gh)1/2 the correct choice will be (b).
5. Solution:A,C
6. sol:- (a)
7. Solution :(c)
8. Solution:(c)

Subjective Question for SHM

Q1. A mass attached to a spring is free to oscillate , with angular velocity ω , in a frictionless horizontal plane. The mass is displaced from it's equilibrium position by a distance x0 towards the center by pushing it with velocity v0 at time t=0. Find the amplitude of resulting oscillations in terms ofω,x0 and v0 .
Ans. A=√[x02+(v022)]


Q2. A uniform cylinder of length l and mass m having crosssectional area Ais suspended , with length vertical , from a fixed point by a massless spring , such that it is half submearged in a liquid of density σ at equilibrium position.. When the cylinder is given a small downward push and released , it starts oscillating verticallywith small amplitude . Calculate the frequency of oscillations of cylinder.
(IIT 1990)
Ans. f=[(k+(σAg)/m]1/2


Q3. What should be the percentage change of length of pendulum in order that clock have same time period when moved from place where g=9.8 m/s2 to another where g=9.81 m/s2.
Ans. .102%

Q4. A 4 kg particle is moving along x axis under the action of the force F=-(π2/16)x N
when t=2 s the particle passes through origin.If x0 is the amplitude of oscillating particle find the equation of elongation.
Ans. x=x0 cos( πt/8 + π/4)


Q5. Two blocks of masses m1 and m2 are connected by a spring and these masses are free to oscillate along the axis of the spring. Find the angular frequency of oscillation.
Ans.
ω=√[k(m1+m2)/m1m2]

Objective Question for SHM

Q 1. Total energy of mass spring system in harmonic motion is E=1/2(mω2A2). Consider another system executing SHM with same amplitude having value of spring constant as half the previous one and mass twice as that of previous one. The energy of second oscillator will be

(a) E
(b) 2E
(c) √ 2E
(d) E/2

Q 2. A particle is executing linear SHM of amplitude A. What fraction of total energy is potential when the displacement is 1/4 times amplitude.

(a) 3/2
(b) 1/16
(c) 1/4
(d) 1/2√ 2

Q 3. Fig below shows two spring mass systems. All the springs are identical having spring constant k and are of negligible mass. If m is the mass of block attached to the spring then the ratio of time period of oscillations of both systems is



(a) 1:2√ 2
(b) 1:1/2√ 2
(c) 1:√ 2
(d) √ 2:1

Q 4. Fig below shows two equal masses of mass m joined by a rope passing over a light pully. First mass is attached to a spring and another end of spring is attached to a rigid support. Neglacting frictional forces total energy of the system when spring is extended by a distance x is



(a) mv2+1/2(Kx2)+mgx
(b) mv2-1/2(Kx2)+mgx
(c) mv2-1/2(Kx2)-mgx
(d) mv2+1/2(Kx2)-mgx

where v = dx/dt , the velocity of mass

Q 5. A spring of force constant k is cut into two pieces such that one piece is four times the length of the other. the longer piece will have force constant equal to

(a) 4k/5
(b) 5k/4
(c) 3k/2
(d) 4k

Q 6. In the system shown below frequency of oscillation when mass is displaced slightely is



(a) f=1/2π(k1k2/(k1+k2)m)1/2
(b) f=1/2π((k1+k2)/m)1/2
(c) f=1/2π(m/(k1k2))1/2
(d) f=1/2π((k1+k2)/(k1k2)m)1/2

Q 7. A simple pendulum is displaced from it's mean position o to a position A such that hight of A above O is 0.05m. It is then released it's velocity when it passes mean position is

(a) .1m/s
(b) 5.0m/s
(c) 1m/s
(d) 1.5m/s

Q 8. A particle is executing SHM at mid point of mean position and extreme position . What is it's KE in terms of total energy E.

(a) E/2
(b) 4E/3
(c) √ 2E
(d) 3E/4

Q 9. A solid cylinder of radius r and mass m is connected to a spring of spring constant k and it slips on a frictionless surface without rolling with angular frequency


(a) √(k/mr)
(b) √(kr/m)
(c) √(k/m)
(d) √(2k/m)


Solutions

Conceptual Question of SHM

1.To execute SHM system must have
a. Elasticity
b. Moment of Inertia
c. Inertia
d. all the above


2. Angular frequency of system executing SHM depends on
a. mass
b. total energy
c.force constant
d. Amplitude


3.A particle of mass m is attached to a massless string of lenght L and is oscillating in vertical plane with one end of string fixed to rigid support.Tension in the string at a certain instant is T=kmg.Then
a. K can never be equal to 1
b. K can never be greater than 1
c. K can never be greater than 3
d K can never be less than 1


4.The bob A of a simple pendulum is released when the string makes an angle 45 with the verical.Its hit another bob B of the same mateial and same mass kept at rest on the table.If the collsion is elastic
a. B moves first and A follows it with half of its intial velocity
b.A comes to rest and B moves with the velocity of A
c Both A and B moves with same velocity of A
d Both A and B comes to rest at B

5.For a particle executing SHM
a.Acceleration is proportional to the displacement in the direction of the motion
b.Acceleration is proportional to the displacement but in opposite direction of the motion
c. Total energy of particle remains constant
d KE and PE of particle remains constant


6. which one of the following statement is true
a. Maximum value of velocity in SHM is A2ω
b.In SHM velocity of the particle is maximum when displacment is maximum
c.Velocity of the particle is zero in SHM when displacement attains its maximum on either side
d.Velocity in SHM vary periodically with time


7. which one of the following statement is true
a. Amplitude and intial displacement of particle in SHM are always equal
b.Amplitude and intial displacement of particle in SHM are never equal
c. Amplitnude of a particle in SHM can be equal to its initial displacement
d. Amplitnude of a particle in SHM can be greater to its initial displacement


8.The amplitutde and phase of a particle executing SHM depends on
a.The displacemnt of particle at t=0
b.The velocity of particle at t=0
c Both Velocity and displacement at t=0
d Neither velocity and displacemnt at t=0

Solutions

Oscillations

PART 2

(1) Some system Executing SHM

a)Oscillations of a Spring mass system

-In this case particle of mass m oscillates under the influence of hooke's law restoring force given by F=-Kx where K is the spring constant

Angular Frequency ω=√(K/m)

Time period T=2π√(m/K)

And frequency is =(1/2π)√(K/m)

Time period of both horizontal ans vertical oscillation are same but spring constant have diffrent value for horizontal and vertical motion

b) Simple pendulum

-Motion of simple pendulum oscillating through small angles is a case of SHM with angular frequency given by
ω=√(g/L)
and Timeperiod
T=2π√(L/g)
Where L is the length of the string.

-Here we notice that period of oscillation is independent of the mass m of the pendulum

c) Compound Pendulum

- Compound pendulum is a rigid body of any shape,capable of oscillating about the horizontal axis passing through it.
-Such a pendulum swinging with small angle executes SHM with the timeperiod

T=2π√(I/mgL)

Where I =Moment of inertia of pendulum about the axis of suspension
L is the lenght of the pendulum

(2) Damped Oscillation

-SHM which continues indefinitely without the loss of the amplitude are called free oscillation or undamped and it is not a real case

- In real physical systems energy of the oscillator gradually decreases with time and oscillator will eventually come to rest.This happens because in acutal physical systems,friction(or damping ) is always present

-The reduction in amplitude or energy of the oscilaltor is called damping and oscillation are call damped

(3) Forced Oscillations and Resonance.

- Oscillations of a system under the influence of an external periodic force are called forced oscillations

- If frequency of externally applied driving force is equal to the natural frequency of the oscillator resonance is said to occur

Oscillations

PART I
-If a particle moves such that it retraces its path regularly after regular interval of time,its motion is said to be periodic Ex-Motion of earth around Sun

-If a body in periodic motion moves back and forth over the same path then the motion is said to be oscillatory motion

-Simple harmonic motion is simplest form of oscillatory motion

-SHM is a kind of motion in which the restoring force is propotional to the displacement from the mean position and opposes its increase.Mathematically restoring force is
F=-Kx
Where K=Force constant
x=displacement of the system from its mean or equilibrium position
Diffrential Equation of SHM is
d2x/dt2 + ω2x=0
Solutions of this equation can both be sine or cosine functions .We conveniently choose
x=Acos(ωt+φ) where A,ω and φ all are constants

-Quantity A is known as amplitude of SHM which is the magnitude of maximum value of displacement on either sides from the equilibrium position

-Time period (T) of SHM the time during which oscillation repeats itself i.e, repeats its one cycle of motion and it is given by
T=2π/ω where ω is the angular frequency

-Frequency of the SHM is the number of the complete oscillation per unit time i.e, frequency is reciprocal of the time period
f=1/T
Thus angular frequncy
ω=2πf

-Velocity of a system executing SHM as a function of time is
v=-ωAsin(ωt+φ)

-Acceleration of particle executing SHM is
a=-ω2Acos(ωt+φ)

So a=-ω2x

This shows that acceleration is proportional to the displacement but in opposite direction

-At any time t KE of system in SHM is
KE=(1/2)mv2
=(1/2)mω2A2sin2(ωt+φ)
which is a function varying periodically in time

-PE of system in SHM at any time t is
PE=(1/2)Kx2
=(1/2)mω2A2cos2(ωt+φ)

-Total Energy in SHM
E=KE+PE
=(1/2)mω2A2
and it remain constant in absense of dissapative forces like frictional forces

Solutions for Kinematics objective

1 a,c,d
Hint:Eliminating t from both the equation,you got trajotory and diffrentiating gives velocity y
x=2t y=2t2 So eliminating gives y=x2/2

2.d
3.d
4.c
5.b,c
6.a,b
7.b
8.b
9.a
10.d

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