11th Class Physics Chapter 7 OSCILLATIONS Short Question Answers

Two characteristics of simple harmonic motion are given as below:-
1.acceleration of a vibrating body is directly proportional to the displacement and is always directed towards the mean (or equilibrium) position.
i.e.      a∞-x
2.Total energy of particle executing SHM remain conserved
3.SHM can be represented by a single harmonic function of sine or cosine in thr from of equation.
x=xo sin(ῳ+ᶲ)
x=x0cos(ῳt+ᶲ)
or
phase (ᶲ) is a measure of how for the oscillator is away from its mean
position at time t=0
The knowledge of phase enable us to find how for from its mean position the oscillation was at  t=0

No, the frequency oscillator is independent of the amplitude of oscillation provided it is small.
As we know that frequency of oscillation of simple pendulum is
F=1/2π∫g/l
This relation shows that frequency does not depend upon the amplitude but it depends upon the length of pendulum and acceleration due to gravity.
Similarly, in the case of mass-spring system , frequency of oscillation of mass is given by
f=1/2π∫k/m
This relation shows that frequency depends upon mass of the body and spring constant ‘k’ but it is independent of the amplitude.

No, we cannot realize an ideal simple pendulum, because for an ideal simple pendulum we must full fill the following conditions : (i) Bob of very small size (points mass) must be used. (ii)The suspension string must be weightless and inextensible (iii) The bob of very small size (point mass) must be suspended from rigid frictionless support. (iv) Air also should be removed from the place of experiment. These conditions could not be fulfilled 100% . Therefore, it is Impossible to realize an ideal simple pendulum in nature.

The total distance travelled by an object moving] With SHM in its time period is equal to 4A,where A is its amplitude. Explanation:- In this question, the total distance in a time equal to period means the distance covered during one complete vibration. In vibratory motion of pendulum. Distance OD=Amplitude=A Distance of half vibration from O to D and back from D to O=2A Distance of half vibration from O to C and back from C to O =2A Total distance covered during one complete vibration =2A+2A=4A. As the time token by one vibration is called time period. Hence, the object moves a total distance equal to 4A in a time equal to its period.

No, the acceleration of harmonic oscillator does not remain constant during its motion.
Given by
aα-x
or      a=-(constant)*x
where x is the displacement from the mean position.
Since the displacement changes continuously during SHM, so its acceleration does not remain constant. The value of acceleration at the mean position will be zero because at this position x=0 and its maximum value will be at the extreme position.

The angle which specifies the displacement as well as the direction of motion of the point executing SHM is called phase angle. Thus, it indicates the state of motion of the vibrating the simple harmonic Oscillor is
Mathematically, the phase angle is Expressed as
θ=ῳt
Where ῳ=angular frequency
t=any instant of time.
(b) It is the angle ‘θ’ which the rotating radius OP makes with reference direction OO1 at any instant ‘t’ as shown in the fag.
It does not define angle between maximum displacement and the driving force.

In order to produce resultant SHM by the addition of two simple harmonic motion, following conditions must be fulfilled
(1)Two SHMs must be parallel (i.e. their phase phases must be in the same direction.
(2)Two SHMs must have the same frequency (i.e. period) but different amplitudes.
(3) These two harmonic motion must have constant phase difference.
If two SHMs are given as
x1=A1 sin ῳt+ and x2=A2 sin(ῳt+ᶲ)
Resultant SHMs will be written as
X=X1+X2=A1 sin ῳt+A2 sin (ῳt+ᶲ)

.(i) y=A sin(ῳt+ᶲ)
This equation represent the displacement of simple harmonic oscillator as a function of time.
Thus, this equation tells that displacement follows a sine curve i.e. varies harmonically.
‘ᶲ’ is initial phase angle which tells us the start of motion. ῳt is the angle subtended in time t with angular frequency ‘ῳ’ starting from initial phase ‘ᶲ’ (ῳt+ᶲ) is the phase angle made with reference direction. ‘y’ is the instantaneous displacement of a particle performing SHM.
‘A’ is the amplitude of the oscillating .
→                 →
(ii) a= -ῳ² x
This  equation represent the variation of acceleration of S.H. oscillator as a function of displacement.
This equation tells that the acceleration of simple harmonic oscillator is directly proportional to its displacement and its directed towards the main position.
In the above equation , ‘a’ is the acceleration of a particle executing SHM.
‘ῳ’ is the angular frequency of the particle.
‘x’ is in instantaneous displacement of an oscillating particle, from the mean position.

. For a body oscillating with SHM, the relation between potential energy , kinetic energy and total energy at any instant is
E ………….. =P.E.+K.E
since total energy of SHM remains constant, therefore any decrease in K,E. or P.E. result increase in P.E. or K.E. respectively.
During  SHM, in the absence of frictional force, the K.E. and P.E, are interchange continuously from one from to another but the total energy remain constant. At mean position, the energy is totally kinetic, i.e, K.E. is maximum but P.E. is zero. At the extreme position the K.E. is completely change into P.E. , i.e. P.E. becomes maximum but K.E. is zero.

. Following are the common phenomena in which resonance. By tuning a dail, the natural frequency of an alternating current in the receiving circuit is made equal to the frequency of the wave broadcast by the desired station. When the two frequency match, energy absorption is maximum and thus transmitted signal becomes large enough due to the resonance. This received signal enables us to hear the programme of desired station (2) Swing A common phenomenon of resonance is provided by pushing a swing. If it is pushed after regular intervals of time (equal to the period of swing, its motion will increase with every push. If the pushes occur at irregular intervals, the swing will hardly vibrate. (3) Microwave oven:- The waves produced in this type of oven have a wave length of 12 cm at a frequency of 2450 MHZ. At this frequency the waves are absorbed due to resonance by water and fat molecules in the food, heating them up and so cooking of food very efficiently and evenly (4)Musical strings;- In the musical string when the frequency of enclosed air column in the wooden boxes under the strings becomes equal to the string frequency ,due to resonance a loud sound of music is heard.

When a mass-spring system is hung vertically and set into oscillations, we see that the amplitude of the oscillatig system becomes smaller with the passage of time. Finally, the oscillations of mass –spring system stop due to friction, air resistance and some other damping force. Thus, mechanical energy of the system is wanted into heat due to the resistive force. Such an oscillator can be called as damped harmonic oscillator.

When oscillations have a large amplitude , then the time period of oscillation depends on the amplitude and its greater. But when the amplitude becomes smaller, the time period becomes nearly independent of amplitude hence becomes constant. So the frequency will remain unaffected (same) if its oscillations die down from large amplitude to small.

Since damping prevents the amplitude from becoming large, therefore damping forces are oftenly used to prevent excessive oscillations is the shock absorber of a car which provides a damping force to prevent excessive oscillations. Thus, damping device are used on machinery in order to protect them from any damage and prolong the life machinery.

. If a singer produces a note of particular natural frequency of glass, then by the process of resonance, their effect on the glass would on increasing. As a result, the amplitude of vibration of glass will become more and more and it is just possible to exceed its elastic limit, thus, the glass may shatter.

In simple harmonic motion (S.H.M.) the acceleration of a body is directly proportional to the displacement . Therefore, it is zero when it is at equilibrium position i.e velocity is maximum at that position. Hence, the acceleration is zero at the equilibrium position when the velocity is greatest . (maximum).

Due to frictional resistance between air and bob, the amplitude of oscillations of the pendulum gradually decreases an it finely stops.

(a) Examples of free oscillation:- (i) A simple pendulum oscillation (vibrates) freely with its natural frequency that depends only upon the length of the pendulum. When it is slight displaced from its mean position. (ii) When a tuning fork is struck against a rubber pad, the prongs being to execute free oscillations. (b) Examples of forced oscillations:- (i) when the free end of the string of simple pendulum is held in hand and is made to oscillate by giving jerks by the hand, the pendulum executes forced oscillations. The frequency of these oscillations is equal to the frequency of oscillation of the force applied by the hand. (ii)The sound board of all stringed musical instruments like sitar, violin etc. execute forced oscillations and the frequency of oscillations is equal to the natural frequency of vibrating string.

(i) Resonance can be used to determine the frequency of a given body. A second body, The natural frequency of which is know is made to act on the given body . If it produces resonance, it is concluded that the given body has the same frequency as the second body. (ii) It is used to determine the speed of sound with resonance apparatuse. (3) It is used to find the natural frequencies of the different bodies. For example, by using a sonometer box we can determine the frequency of a tuning fork with the phenomenon of resonance produced in the string of the box.

The basic conditions for a system to execute S.H.M are (1) The system must have inertia (2) There must be an elastic restoring force acting on the system (3) The system must obey Hook’s law (4) The acceleration of the system should be proportional to the displacement (from the main position) and must always be directed towards the mean position.

Frequency of a second, s pendulum:-
for a second pendulum
Time period =T= 2sec
But the relation between the frequency and the time period is given by
f= 1/T
Therefore, the frequency of a second’s pendulum is given by
F= ½=0.5 vibrations/sec
Or        F=05vib-s^-1

Example of S.H.M:- (1) The spring-mass system (2) Oscillation of simple pendulum (3) Motion of a swing (4) The up and down motion of a loaded elastic string. (5) Motion of pendulum of clock

Uses of simple pendulum
(1) The value of ‘g’ can be found by simple pendulum, because both T and l can be directly measured.
(2) The height of a tower can be measured by determining the time period of a pendulum suspend to the top of that tower up to the ground. i.e. l=gT²/4π²
(3) We can find the frequency ‘f’ of a vibrating body by simple pendulum.

Uses of simple pendulum (1) The value of ‘g’ can be found by simple pendulum, because both T and l can be directly measured. (2) The height of a tower can be measured by determining the time period of a pendulum suspend to the top of that tower up to the ground. i.e. l=gT2/4π2 (3) We can find the frequency ‘f’ of a vibrating body by simple pendulum.

A small heavy metallic bob (or mass) suspended from frictionless support by a light and inextensible string fixed at its upper end. Is called simple pendulum. This distance between the point of suspension and the center of bob is called the length of the pendulum.

The angle which specifies the displacement as well as the direction of motion \of the point executing SHM is known as phase. It is given by θ= ῳt In other words, the phase determines the state of motion of the vibrating point. The phase angle is not the angle between maximum displacement and the driving force.

The phenomenon of resonance occurs when the frequency of the applied is equal to one of the natural frequencies of vibration of the forced (driven) harmonic oscillator. Examples:- (1) Tuning of radio is an example of electrical resonance. (2) A swing is an example of mechanical resonance (3)Heating and cooking of food by microwave oven (4)The soldiers are ordered to break, their steps while crossing a big bridge, because if the frequency of their steps coincides with the natural frequency of the bridge, the bridge may be set into vibration of large amplitude. Thus, the bridge may collapse due resonance.

A physical system undergoing forced vibration is known as driven (forced) harmonic oscillator.

Such type of oscillations, in which the amplitude decreases steadily with time are called damped oscillations. Its Applications An application of damped oscillations is the shock absorber of a car which provides a damping force to prevent excessive oscillations. If the shock absorber, s are defective then the car becomes bouncy and uncomfortable.

Restoring force is defined as that force which tends to move the body back to its original position when the applied force is removed.
Mathematically, is can be written as
F_r= -kx
Negative sign shows that the force ‘f’ is negative i.e directed opposite to displacement  ‘x’ (towards mean position).

Force =F10 N
Displacement =X=0.2m
Spring constant = k
Using the formula, F=kx
Or      k= f/x=10/0.2
=100/2=50 Nm^-2
Hence, K=50 Nm^-1  Ans

Damping is a process in which energy is dissipated (wasted) from the oscillating system. Damping prevents the amplitude from becoming excessively large.

When a body oscillates with the natural frequency without the interference of an external force, then it is said to be performing free vibrations (or oscillations) For example, a simple pendulum vibrates freely with its natural frequency.

If a free oscillating system is forced to vibrate under an external force, force vibrations will be produced which are known as forced oscillations. For example, if the mass (or bob)of vibrating pendulum is struck again , then forced vibration are produced. The other example of forced vibration is that the vibration of a factory floor caused by the running of heavy machinery are called forced vibration.

When we turn the knob of a radio, to tune a station, we actually change the natural frequency of the electric circuit of the receiver to make it equal to the transmission frequency of the radio station. When the two frequencies become equal (coincide) resonance tekesplace and we hear the program of desired station.

. It states that total energy of the vibrating mass and spring remains constant at every instant in its path. This is called as the law of conservation of energy in SHM. Explain of simple pendulum:- In the case of vibrating simple pendulum, gravitational P.E. is conserved into K.E. at the mean position. The K.E is converted into P.E. as the bob reaches to the top of the swing. Thus, the total energy of bob remain the same (constant). Due to frictional forces energy is wasted in the form of heat energy, so the system does not vibrate indefinitely.

For a given spring constant, force constant is given by K=f/x. By cutting the spring into two equal halves, the length of each spring is halved and extension in each half under a given load will also be halved. Thus, K=f/x/2=2(f/x)=2K Hence, the spring constant of each half becomes double the constant of the original spring.