# SIMPLE HARMONIC MOTION AND WAVES.what are harmonic motion?

Q.1. what’s straightforward harmonic motion? What square measure the required conditions for a body to execute straightforward harmonic motion?

Ans. straightforward periodic motion (SHM)

To and fro periodic motion within which acceleration of the body is directly proportional to the displacement of the body from the mean position and is-always directed towards the mean position is understood as straightforwardperiodic motion.

Conditions for straightforward periodic motion

A body should fulfil the subsequent 2 conditions so as to execute straightforward harmonic motion:

Acceleration of the body is directly proportional to its displacement.
Acceleration of the body is often directed towards the mean position. Mathematically, each conditions are oftenwritten as: a o c – x
Q.2. Prove that vibrating motion of a mass connected to a spring is SHM.

Ans. think about a mass connected with a spring. If the spring is stretched or compressed through alittledisplacement ‘x’ from its mean position, it exerts a force F on the mass.

According to the Hooke’s law, this force is directly proportion to the amendment long ‘x’ of the spring.

F = kx …… (1)

where k may be a constant referred to as the spring constant.

The value of k are often obtained from Equation (1) as:” ‘ ‘

The spring constant k is outlined as:

The quantitative relation of the force engaged on the spring to the rise in its length is named the spring constant .

Unit: The SI unit of spring constant is newton per metre (Nm~1).

When the spring is force by applying a force F, the length of the spring will increasewhen cathartic this force, spring moves towards the mean position. throughout the motion, if the displacement of the mass is ‘x’ then

F = -k x …… (2)

The negative sing means the force exerted by the spring is often directed opposite to the displacement of the mass. as a result of the spring force invariably acts towards the mean position, it’s referred to as a restoring force, outlined as:

A restoring force invariably pushes or pulls the article activity periodic motion towards the mean position.

Using Newton’s second law

F = ma

or a=£

Putting the worth of F from equivalent weight. (2), we get

-k x a=~m

Since k/m is constant, so a«: -x

This means that the acceleration of the body is directly proportional to its displacement from the mean position and is often directed towards the mean position. This shows that motion of a mass connected to a spring is straightforward periodic motion.

Q.3. will a ball placed during a bowl executes SHM. Explain.

Ans. once the ball is at the mean position O i.e. at the centre of the bowl, web force engaged on the ball is zero. currently if we tend to bring the ball at position A and so unharness it, the ball can begin moving towards the mean position O and therefore the extreme position B thanks to the restoring force caused by its weight. The ball continues its motion concerning the mean position O. This to and fro motion concerning the mean position O continues until all the energy is lost thanks to friction. Thus, to and fro motion of the ball a few fastened position placed during a bowl is SHM.

Q.4. Show that motion of a straightforward setup is SHM.

Ans. a straightforward setup consists of alittle bob of mass ‘m’ suspended from a light-weight string of length L fastened at its higher finish with a support. once the bob is at the mean position O, world wide web force engaged on the bob is zero and therefore the bob is stationary. currently if we tend to bring the bob to the intense position A, world wide web force engaged on the bob isn’t zero. because the part of the load of the bob mass G is balanced by the strain within the string, therefore, there’s no force acting on the string.

Hence, there’s no motion on this direction. The part of weight mass G is directed towards the mean position and acts as a restoring force. thanks to this force, once the bob is free from purpose A, it starts moving towards the mean position O. At the mean position O, the speed of the bob is most and thanks to inertia, the bob doesn’t stop at purpose O rather it continues to maneuver towards the intense position B. the speed of the bob becomes zero on reaching at purpose B.

On reaching at purpose B, the bob starts moving

towards the mean position O thanks to the restoring force mg sin half dozenduring this manner, the bob continues its to and fro motion between the points A and B concerning the mean position O.

It is clear from the on top of discussion that acceleration of the bob is often directed towards the mean position O. thence the motion of the easy setup is SHM.

Q.5. At that positions, the P.E. of a straightforward setup has minimum and most values?

Ans. throughout its to and fro the P.E. of a straightforward setup is most at its extreme positions and it’s minimum at its mean position.

Q.6. At that positions, the K.E. of a straightforward setup has minimum and most values?

Ans. throughout its to and fro motion, the K.E. of a straightforward setup is minimum at its extreme positions and it’s most at its mean position.

Q.7. What precautions ought to be unbroken in mind, whereas crucial the period of a straightforward pendulum?

Ans. so as to work out the period of a straightforward setup, the amplitude of its vibration ought to be unbrokentinyas an example, if the length of the setup is one’ metre, then its amplitude of vibration ought to be unbrokenten cm.

Q.8. Write equations for period of a mass-spring system and a straightforward setup.

Ans. The period T of the SHM of a mass connected to a spring is given by the subsequent equation:

The period T of the SHM for a straightforward setup is given by the subsequent equation:

Q.9. What square measure the characteristics or vital options of SHM?

Ans. The characteristics of SHM are:

(i) A body corporal punishment SHM invariably vibrates a few fastened position.

(ii) The acceleration of a body corporal punishment SHM is ‘always directed towards the mean position.

(iii) The magnitude of acceleration is often directly proportional to its displacement from the mean position i.e., acceleration are going to be zero at the mean position whereas it’ll be most at the intense positions.

(iv) the speed of a body corporal punishment SHM is most at the mean position and nil on the intense positions.

Q.10. suppose 6f many examples in existence of motion that square measurestraightforward harmonic.

Ans. Some samples of SHM are:

The motion of a mass connected to a spring.
The motion of a straightforward setup.
The motion of ball during a bowl.
Up and down motion of a leaf in water pool.
Motion of a plucked string fastened at each its ends.
Q.11. outline the subsequent terms:

(i) vibrations

(ii) period

(iii) Frequency

(iv) Amplitude

(v) Wave length

Ans. Vibration

One complete trip of a moving body concerning its mean position is named one vibration.

Time amount

The time taken by a moving body to complete one vibration is named period. ‘ .

Unit: The SI unit of your time amount is second(s).

Frequency

The number of vibrations per cycles of a moving body in one second is named frequency.

Unit: The SI unit of frequency is hertz (Hz).

Amplitude

The maximum displacement of a moving body on either aspect from its mean position is named amplitude.

Unit: The SI unit of a amplitude is metre (m).

Wavelength

The distance between 2 consecutive crests or troughs is. referred to as wavelength.

It is denoted by X.

Unit: The SI unit of wavelength is meter (m).

Q.12. What square measure damped oscillations? however damping increasingly reduces the amplitude of oscillation?

Ans. DAMPED OSCILLATIONS

The oscillations of a system within the presence of some resistive force square measure referred to as damped oscillations.

Practically, altogether systems, the force of friction retards the motion, therefore the system doesn’t oscillate indefinitely. The friction reduces the energy of the system as time passes, and therefore the motion is claimed to be damped. This damping increasingly reduces the amplitude of the motion as shown within the figure.

Q.13. Write a note on one use of damped motion.

Ans. Shock absorbers in cars square measure one use of damped motion. A muffler consists of a piston moving through a liquid like oil. The higher a part of the muffler is firmly connected to the body of the automobileonce the automobile travels over a bump on the road, the automobile could vibrate violently. The shock absorbers damp these vibrations and convert their energy into energy of the oil.

Q.14. what’s a wave? however square measure waves produced?

Ans. WAVE

A wave may be a disturbance created within the medium.

PRODUCTION OF WAVES

Waves square measure created during a medium thanks to such a disturbance that causes the particles of the medium to perform vibrating or periodic motion. during this manner, this disturbance is transmitted from one a part of the medium to the opposite within the kind of waves.

Q.15. Elaborate the distinction between the mechanical and magnetism waves. conjointly offer samples of every,

Ans. MECHANICAL WAVES

Such waves that need any medium for his or her propagation square measure referred to as mechanical waves.

Examples

(i) Water wave

(ii) Sound waves

(iii) Waves created in springs

(iv) Waves created in strings

ELECTROMAGNETIC WAVES

Such waves that don’t need any medium for his or her propagation square measure referred to as magnetismwaves.

Examples

(ii) tv waves

(iii) X-rays

(iv) Heat and lightweight waves

Q.16. What square measure longitudinal waves? however square measure they produced?

Ans. LONGITUDINAL WAVES

In longitudinal waves, the particles of the medium pull back and forth on the direction of propagation of the wave.

PRODUCTION OF LONGITUDINAL WAVES

Longitudinal waves are often created on a spring (slinky) placed on a sleek floor or a protracted bench. Fix one finish of the slinky with a rigid support and hold another finish into your hand. currently provides it an everydaypush and pull quickly within the direction of its length as shown within the figure.

A series of disturbances can begin moving on the length of the slinky, Such a wave consists of regions referred to as compressions and rarefactions. The compressions and rarefactions pull back and forth on the direction of motion of the waves.

Examples: Waves created in springs square measure longitudinal waves.

Q.17. What square measure cross waves? however square measure they produced?

Ans. cross WAVES

In cross waves, the particles of the medium vibrate perpendicular to the

direction of propagation of the wave.

PRODUCTION OF cross WAVES

Stretch out a slinky on a sleek floor or a bench with one finish fastened. Hold the opposite finish of the slinky and move it up and down quickly. A wave within the kind of crests and troughs can begin traveling towards the fastened finish. The crests and troughs move perpendicular to the direction of propagation of the wave.

Examples

(i) Waves created on the string.

(ii) Waves turn out on the surface of water.

(iii) light-weight waves.

Q.18. outline the subsequent terms:

(i) Crest

(II)Trough

(iii) Compression

(iv) Rare faction.

Ans. Crest

The crest is that the highest purpose of the particles of the medium from the mean position.

Trough

A trough is that the lowest purpose of the particles of the medium from the mean position.

Compression

The region wherever the particles of the medium square measure closed along is understood as compression.

Rarefaction

The region wherever the particles of the medium square measure spaced apart is understood as concentration.

Q.19. Draw a cross wave with associate degree amplitude of two cm and a wavelength of four cm. Label a crest and trough on the wave.

Ans:

Q.20. what’s wave equation? Derive a relationship between speed, frequency and wavelength of a wave?

Ans. differential equation

The relationship between the speed, frequency and wavelength of the wave is understood as differential equation.

A relation between speed of a wave, distance coated by the wave and time taken is written as:

Distance

or d t

If time taken by the wave in moving .from one purpose to a different is capable the period, then the space coatedby the wave are going to be capable one wavelength. Hence Eq. (i) are often written as:

or

…… (ii)

But period T is that the reciprocal of the frequency f i.e., T = (Hi)

By golf shot the worth of T in Equation (ii), we tend to get v = f X ….(1)

Equation (1) is named the differential equation.

Q.21. what number strategies square measure there for the transfer of energy from one place to another?

Ans. There square measure 2 strategies for the transfer of energy from one place to a different.

Transfer of energy through matter.
Transfer of energy through waves.
Q.22. Waves square measure the means that of energy transfer while not transfer of matter. Justify this statement with the assistance of a straightforward experiment?

Ans. WAVES AS CARRIES OF ENERGY

Waves square measure the means that of energy transfer while not transfer of matter. allow us to prove it with the assistance of associate degree experiment.

EXPERIMENT

Drop a stone into a pool of water. Water waves are going to be created on the surface of water and can travel outward. Place a cork at a ways from the falling stone. once the waves reach the cork, it’ll move up and down onthe motion of the water particles by obtaining energy from the wave.

This experiment shows that water waves like alternative waves transfer energy from one place to the oppositewhile not transferring matter.

Q.23. offer some examples within which energy is transferred from one place to a different through matter.

Ans. the subsequent examples justify however energy is transferred through matter:

1.When a gun is discharged, the bullet gets energy through explosion and moves towards the target. during thisexample the bullet contains the mechanical energy.

The energy of running water within the tunnel of a dam is additionally kinetic in nature.
Q.24. what’s ripple tank? however square measure waves created on the surface of water with the assistance of ripple tank apparatus?

Ans. RIPPLE TANK

Ripple tank may be a device to supply water waves and to review their characteristics.

This equipment consists of an oblong receptacle having glass bottom and placed nearly [*fr1] metre on top of the surface of a table. Waves are often created on the

surface of water gift within the receptacle by means that of a vibrator.

This vibrator is associate degree .oscillating motor fastened on a picket plate over the receptacle such its lower surface simply touches the surface of water. On setting the vibrator ON, this picket plate starts moving to come up with plane water waves. an electrical bulb is decorated on top of the receptacle to observer the image of water waves on the paper or screen.

Q.25. justify the development of reflection of waves with the assistance of ripple tank equipment.

Ans. REFLECTION OF WAVES

When waves taking possession one medium fall on the surface of another medium, they recover into the primarymedium such the angle of incidence is capable the angle of reflection. This development is named reflection of wave.

Now, we tend to justify the reflection of waves with the assistance of ripple tank.

Place a bar within the ripple tank, the mirrored wave are often seen to adapt the law of reflection. The ripples canreplicate from the bar. If the bar is placed at associate degree angle to the wave front i.e. the angle of the incident wave on the conventional are going to be capable the angle of mirrored wave

Q.26. justify refraction of waves with the assistance of ripple tank equipment.

Ans. REFRACTION OF WAVES

When waves from one medium enter within the second medium at some angle their direction of travel couldamendment. This development is named refraction of waves.

The speed of a wave in water depends on the depth of water. If a block is submerged within the ripple tank, the depth of water within the tank are going to be shallower over the block than elsewhere. once water waves enter the region of shallow water, their wavelength decreases .But the frequency of the water waves remains an equivalent in each components of water, as a result of it’s capable the frequency of the vibrator.

For the observation of refraction of water waves, we tend to repeat this experiment such the boundary between the deep and therefore the shallower water is at associate degree angle to the wave front. currently we are going to observe that additionally to the amendment in wavelength, the waves changes their direction of propagation also The direction of propagation is often traditional to the wave fronts. this modification of depth of water waves whereas passing from a district of trouble thereto of shallower is named refraction.

Q.27. On that issue speed of water waves depends?

Ans. The speed of water waves depends upon the depth of water.

The speed of water waves is reduced once they enter from deep to shallow water.

The speed of water waves is magnified once they enter from shallow to deep

water.

Q.28. justify the method of generation of optical phenomenon of water waves victimization ripple tank equipment.

Ans. optical phenomenon

The bending or spreading of waves round the sharp edges or comers of obstacles is named optical phenomenon.

To observer the development of optical phenomenon of water waves, generate plane waves during a ripple tank and place 2 obstacles in line in such the way that separation between them is capable the wavelength of water waves. when passing through alittle slit between the 2 obstacles, the wave can unfold in each direction and alterinto virtually curving pattern (Fig. 10.13). optical phenomenon of waves will solely be discovered if the dimensionsof the obstacle is comparable the wavelength of the wave. Figure 10.14 show the optical phenomenon of waves whereas passing a slit with size larger than the wavelength of the wave. solely alittle optical phenomenonhappens close to the comers of the obstacle.

Q.29. will increasing the frequency of a wave conjointly will increase its wave length? If not, however square measure these quantities related?

Ans. On increasing the frequency of a wave, the wavelength of the wave doesn’t increase; however it decrease. Thus, frequency and wavelength square measure reciprocally proportional to every alternative.

SOLVED data

Q.1 what’s the displacement of associate degree object in SHM once the kinetic and potential energies square measure equal?

Ans. mechanical energy and P.E. square measure most at mean and extreme positions severallywhereverdisplacement of the body corporal punishment SHM is zero and most severally.

Q.2 Tell whether or not or not these motions square measure samples of straightforward harmonic motion:

(a) up and down motion of a leaf in water pool.

(b) motion of a ceiling fan.

(c) motion of hands of clock.

(d) motion of a plucked string fastened at each its finish.

(e) movement of honey bee.

Ans. Among of these motions, solely example of SHM is that the up and down motion of a leaf in water pool a fewfastened position.

Q.3 Do mechanical waves taste vacuum, that is, empty space?

Ans. Mechanical waves need some material medium for his or her propagation, and thence they can not tasteempty area i.e. vacuum.

Q.4 What do the dark and bright fringes .on the screen of ripple tank represents?

Ans. The dark and bright fringes on the screen of ripple tank represent troughs and crests severally.

Q.5 What happens to the angle of refraction once water waves pass from deep to shallow a part of water?

Ans. once water waves travel from deep to shallow water, angle of refraction decreases. It is .due to the decrease within the speed of water waves within the shallow water.

Q.6 Do the magnitude of angle of incidence and angle of refraction equal?

Ans. Angle of incidence and angle of refraction don’t seem to be equal thanks to totally different depth of water in shallow and deep components.

SUMMARY

The to and fro periodic motion within which acceleration of the body is directly proportional to the displacement of the body from the mean position and is often directed towards the mean position is named straightforwardperiodic motion.

The motion of a mass connected to a spring, pendulum which of a ball within a bowl square measure samples ofstraightforward periodic motion.

Time -taken by the easy setup to complete one cycle is named its period.

Time period depends upon the length of the setup and is freelance of the mass and amplitude of the setup.

The number of cycles completed in one second is named frequency of the setup.

The maximum displacement from the mean position of a body activity SHM is named amplitude.

The process of transferring energy from one place to a different while not the transfer of matter, is understood as wave.

Those waves that need some medium for his or her propagation square measure referred to as mechanical waves:

Electromagnetic waves don’t need any medium for his or her propagation.

Transverse waves square measure the mechanical waves within which particles of the medium vibrate concerningtheir mean position perpendicular to the direction of propagation of the waves.

Compressional (longitudinal) waves square measure the mechanical waves within which particles of the medium vibrate concerning their mean position parallel to the direction of propagation of the waves.

The speed (v) of a wave is capable the merchandise of frequency (f) and wavelength (A.) i.e., v = f X

Ripple tank may be a device wont to turn out water waves and to demonstrate totally different properties of water waves like reflection, refraction and optical phenomenon.

When a wave traveling from one medium falls on the surface of another medium, it’s going to recover into the primary medium. This development is named reflection of waves.

When waves from one medium enter the second medium at some angle, their direction of travel couldamendment. This phenomenor is named refraction of waves.
The bending of waves around obstacles or sharped, is named optical phenomenon of waves.