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PHYSICS

By N.K.C. SIR

MECHANICAL W AV E S CONTENTS KEY CONCEPTS ................................................................... Page –2 OBJECTIVE QUESTION BANK ......................................... Page –5 SUBJECTIVE-I ...................................................................... Page –31 SUBJECTIVE-II ..................................................................... Page –35 ANSWER KEY ....................................................................... Page –38

KEY CONCEPTS 1.

Wave Equation :

(i)

The equation for a progressive wave travelling in the positive x-direction is  t x y = sin 2 π  −  , T λ

where y is the displacemnet at point x, at time t, A is the amplitude, T is the period and λ is the wavelength. The frequency is

(ii)

1 λ and the velocity of the wave is . T T

The velocity of propagation of a transverse wave in a streched string

T m

v=

where T is the tension in the string and m is the mass per unit length of the string. (iii)

The power transmitted along the string is proportional to the square of the square of the amplitude and square of the frequency of the wave. 1 ω2 A 2 F Pav = = 2π2 µvA2v2. 2 v

(iv)

The equation for a stationary wave is 2πx   2πt  sin y =  2A cos λ  T 

(v)

In a sonometer wire of length L and mass per unit length m under tension T vibrating in n loops fn=

n 2L

T m

(vi)

Pitch, loudness and quality are the characteristics of a musical note. Pitch depends on the frequency. Loudness depends on intensity and quality depends on the waveform of the constituent overtones.

(vii)

Resonance occurs when the forcing frequency is equal to the natural frequency of a vibrating body.

(viii)

Velocity of propagation of sound in a gas =

γP , where D is the density of the gas and γ is the ratio of D

specific heats.

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2.

Vibrating air columns :

(i)

In a pipe of length L closed at one end, the funamental note has a frequency f1 =

v , where v is the 4L

velocity of sound in air. v = 2f1 L

(ii)

The first overtone f2 =

3. (i)

Propagation of sound in solids : The velocity of propagation of a longitudinal wave in a rod of Young’s modulus Y and density ρ is given by v=

Y ρ

(ii)

In a sonometer wire of length L and mass per unit length m under tension T vibrating in n loops

(iii)

n T 2L m Propagation of sound in gases

fn=

Laplace formula v =

γP ρ

where γ is the ratio of specific heats, P is the pressure and ρ is the density.

vt T = = v0 T0 4. (i)

273 + t 273

Doppler Effect : When a source of sound moves with a velocity vs in a certain direction, the wavelength decreases in front of the source and increases behind the source. λ’ (in front) =

v − vs v v = fs ; f’ = fs λ' v − vs

λ’’(behind) =

v v v + vs = fs ; f’’ = λ ' ' v + vs fs

Here v is the velocity of sound in air. (ii) (a)

v − v0 fs v When the source is moving towards the observer and the observer is moving away from the source, the apparent frequency

The apparent frequency =

v − v0 fa = v − v f s s

(b)

When the source and the observer are moving towards each other. v + v0 fa = v − v f s s

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(c)

When the source and observer are moving away from each other, v − v0 fa = v + v f s s

(d)

When the source is moving away from the observer and the observer is moving towards the source v + v0 fa = v + v f s s

Here all velocities are relation to the medium. 5.

Loudness of sound : The loudness level B of sound is expressed in decibels, I B = 10 log I

0

where I is the intensity, I0 is a reference intensity. 6.

Beats : When two tuning forks of close but different frequencies f1 and f2 are vibrating simultaneously at nearby places, a listener observes a fluctuation in the intensity of sound, called beats. The number of beats heard per second is f1 – f2.

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OBJECTIVE QUESTION BANK ONLY ONE OPTION IS CORRECT. Take approx. 2 minutes for answering each question. Q.1

A wave is propagating along x-axis. The displacement of particles of the medium in z-direction at t = 0 is given by: z = exp[ –(x + 2)2] , where ‘x’ is in meters. At t = 1s, the same wave disturbance is given by: z = exp[ – (2 – x)2 ]. Then, the wave propagation velocity is (A) 4 m/s in + x direction (B) 4 m/s in –x direction (C) 2 m/s in + x direction (D) 2 m/s in – x direction

Ans.

Q.2

Figure shown the shape of part of a long string in which transverse waves are produced by attaching one end of the string to tuning fork of frequency 250 Hz. What is the velocity of the waves? (A) 1.0 ms–1 (B) 1.5 ms–1 –1 (C) 2.0 ms (D) 2.5 ms–1

Ans.

Q.3

A block of mass 1 kg is hanging vertically from a string of length 1 m and mass/length = 0.001 Kg/m. A small pulse is generated at its lower end. The pulse reaches the top end in approximately (A) 0.2 sec (B) 0.1 sec (C) 0.02 sec (D) 0.01 sec

Ans.

Q.4

A uniform rope having some mass hanges vertically from a rigid support. A transverse wave pulse is produced at the lower end. The speed (v) of the wave pulse varies with height (h) from the lower end as:

(A)

(B)

(C)

(D)

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Ans.

Q.5

Consider a function y = 10sin2(100πt+5πz) where y, z are in cm and t is in second. (A) the function represents a travelling, periodic wave propagating in (−z) direction with speed 20m/s. (B) the function does not represent a travelling wave. (C) the amplitude of the wave is 5 cm. (D) the amplitude of the wave is 10 cm.

Ans.

Q.6

The displacement from the position of equilibrium of a point 4 cm from a source of sinusoidal oscillations is half the amplitude at the moment t = T/6(T is the time period). Assume that the source was at mean position at t = 0. The wavelength of the running wave is (A) 0.96 m (B) 0.48 m (C) 0.24 m (D) 0.12 m

Ans.

Q.7

The period of oscillations of a point is 0.04 sec. and the velocity of propagation of ocillation is 300m/sec. The difference of phases between the ocillations of two points at distances 10 and 16m respectively from the source of ocillations is (A) 2π (B) π/2 (C) π/4 (D) π

Ans.

Q.8

A motion is described by y =

3 where y, x are in meter and t is in second. a + ( x + 3t ) 2 2

(A) This represents equation of progressive wave propagating along –x direction with 3 ms–1. (B) This represents equation of progressive wave propagating along +x direction with 3 ms–1. (C) This does not represent a progressive wave equation. (D) data is insufficient to arrive at any conclusion. ETOOS Academy Ltd. : F-106, Road no.2, Indraprastha Industrial Area, End of Evergreen Motors (Mahindra Showroom), BSNL Office Lane, Jhalawar Road, Kota, Rajasthan (324005) [6]

Ans.

Q.9

A pulse shown here is reflected from the rigid wall A and then from free end B. The shape of the string after these 2 reflection will be (A)

(B)

(C)

(D)

Ans.

Q.10

A wave pulse on a string has the dimension shown in figure. The waves speed is v = 1 cm/s. If point O is a free end. The shape of wave at time t = 3 s is :

(A)

(B)

(C)

(D)

Ans.

Q.11

A string 1m long is drawn by a 300Hz vibrator attached to its end. The string vibrates in 3 segments. The speed of transverse waves in the string is equal to (A) 100 m/s (B) 200 m/s (C) 300 m/s (D) 400 m/s

Ans.

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Q.12

A wave is represented by the equation y = 10 sin 2π(100t−0.02x) + 10 sin 2π(100t+0.02x). The maximum amplitude and loop length are respectively (A) 20 units and 30 units (B) 20 units and 25 units (C) 30 units and 20 units (D) 25 units and 20 units

Ans.

Q.13

The resultant amplitude due to superposition of two waves y 1 = 5sin (wt − kx) and y2 = −5 cos (wt − kx − 150°) (A) 5

(B) 5 3

(C) 5 2 − 3

(D) 5 2 + 3

Ans.

Q.14

A wave represented by the equation y = A cos (kx – ωt) is superimposed with another wave to form a statioary wave such that the point x =0 is a node. The equation of the other wave is: (A) –A sin (kx + ωt) (B) – A cos (kx + ωt) (C) A sin (kx + ωt) (D) A cos (kx + ωt)

Ans.

Q.15

A taut string at both ends vibrates in its nth overtone. The distance between adjacent Node and Antinode is found to be 'd'. If the length of the string is L, then (A) L = 2d (n + 1) (B) L = d (n + 1) (C) L = 2dn (D) L = 2d (n – 1)

Ans.

Q.16

 20  A standing wave y = A sin  π x cos (1000πt) is maintained in a taut string where y and x are 3 expressed in meters. The distance between the successive points oscillating with the amplitude A/2 across a node is equal to (A) 2.5cm (B) 25cm (C) 5cm (D) 10cm

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Ans.

Q.17

A string of length 1m and linear mass density 0.01kgm−1 is stretched to a tension of 100N. When both ends of the string are fixed, the three lowest frequencies for standing wave are f1, f2 and f3. When only one end of the string is fixed, the three lowest frequencies for standing wave are n1, n2 and n3. Then (A) n3 = 5n1 = f3 = 125 Hz (B) f3 = 5f1 = n2 = 125 Hz (C) f3 = n2 = 3f1 = 150 Hz

(D) n2 =

f1 + f 2 2

= 75 Hz

Ans.

Q.18

A sonometer wire has a total length of 1 m between the fixed ends. Two wooden bridges are placed below the wire at a distance 1/7 m from one end and 4/7 m from the other end. The three segments of the wire have their fundamental frequencies in the ratio : (A) 1 : 2 : 3 (B) 4 : 2 : 1 (C) 1 : 1/2 : 1/3 (D) 1 : 1 : 1

Ans.

Q.19

A person can hear frequencies only upto 10 kHz. A steel piano wire 50 cm long of mass 5 g is stretched with a tension of 400 N. The number of the highest overtone of the sound produced by this piano wire that the person can hear is (A) 48 (B) 50 (C) 49 (D) 51

Ans.

Q.20

A chord attached to a vibrating form divides it into 6 loops, when its tension is 36 N. The tension at which it will vibrate in 4 loops is (A) 24 N (B) 36 N (C) 64 N (D) 81 N

Ans.

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Q.21

A string of length 0.4 m & mass 10−2 kg is tightly clamped at its ends . The tension in the string is 1.6 N. Identical wave pulses are produced at one end at equal intervals of time, ∆ t . The minimum value of ∆ t which allows constructive interference between successive pulses is : (A) 0.05 s (B) 0.10 s (C) 0.20 s (D) 0.40 s

Ans.

Q.22

Figure, shows a stationary wave between two fixed points P and Q. Which point(s) of 1, 2 and 3 are in phase with the point X? (A) 1, 2 and 3 (B) 1 and 2 only (C) 2 and 3 only (D) 3 only

Ans.

Q.23

A wave travels uniformly in all directions from a point source in an isotropic medium. The displacement of the medium at any point at a distance r from the source may be represented by (A is a constant representing strength of source) (A) [A/ r ] sin (kr – ωt) (C) [Ar] sin (kr – ωt)

(B) [A/r] sin (kr – ωt) (D) [A/r2] sin (kr – ωt)

Ans.

Q.24

Three coherent waves of equal frequencies having amplitude 10 µm, 4µm and 7 µm respectively, arrive at a given point with successive phase difference of π/2. The amplitude of the resulting wave in µm is given by (A) 5 (B) 6 (C) 3 (D) 4

Ans.

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Q.25

A person standing at a distance of 6 m from a source of sound receives sound wave in two ways, one directly from the source and other after reflection from a rigid boundary as shown in the figure. The maximum wavelength for which, the person will receive maximum sound intensity, is (A) 4 m

(B)

16 m 3

(C) 2 m

(D)

8 m 3

Ans.

Q.26

The ratio of intensities between two coherent soud sources is 4 : 1. The differenmce of loudness in dB between maximum and minimum intensities when they interfere in space is: (A) 10 log 2 (B) 20 log 3 (C) 10 log 3 (D) 20 log 2

Ans.

Q.27

In Quincke’s tube a detector detects minimum intensity. Now one of the tube is displaced by 5 cm. During displacement detector detects maximum intensity 10 times, then finally a minimum intensity (when displacement is complete). The wavelength of sound is: (A) 10/9 cm (B) 1 cm (C) 1/2 cm (D) 5/9 cm

Ans.

Q.28

The ratio of maximum to minimum intensity due to superposition of two waves is the intensity of component waves is 25 16 (B) (A) 25 4

(C)

4 49

(D)

49 . Then the ratio of 9

9 49

Ans.

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Q.29 Two waves of sound having intensities I and 4I interfere to produce interference pattern. The phase difference between the waves is π/2 at point A and π at point B. Then the difference between the resultant intensities at A and B is (A) 2I (B) 4I (C) 5I (D) 7I Ans.

Q.30

Sound waves of frequency 660 Hz fall normally on a perfectly reflecting wall. The shortest distance from the wall at which the air particle has maximum amplitude of vibration is (velocity of sound in air is 330 m/s) (A) 0.125 m (B) 0.5 m (C) 0.25 m (D) 2 m

Ans.

Q.31

An open organ pipe of length L vibrates in second harmonic mode. The pressure vibration is maximum (A) at the two ends (B) at a distance L/4 from either end inside the tube (C) at the mid-point of the tube (D) none of these

Ans.

Q.32

An open organ pipe of length l is sounded together with another organ pipe of length l + x in their fundamental tones (x 0) x in meter and t in second travelling in a uniform string. The pulse passes through a boundary beyond which its velocity becomes 2.5 m/s. What will be the amplitude of pulse in this medium after transmission?

Ans.

Q.7

A 40 cm long wire having a mass 3.2 gm and area of c.s. 1 mm2 is stretched between the support 40.05 cm apart. In its fundamental mode. It vibrate with a frequency 1000/64 Hz. Find the young’s modulus of the wire.

Ans.

Q.8

A string of mass 0.2kg/m in length L=0.6m is fixed at both ends and stretched such that it has a tension of 80N. The string is vibrating in its third normal mode, has an amplitude of 0.5cm. What is the frequency of oscillation? What is the maximum transverse velocity amplitude?

Ans.

Q.9

A stretched uniform wire of a sonometer between two fixed knife edges, when vibrates in its second harmonic gives 1 beat per second with a vibrating tuning fork of frequency 200 Hz. Find the percentage change in the tension of the wire to be in unison with the tuning fork.

Ans.

Q.10

A string fixed at both ends has consecutive standing wave modes for which the distances between adjacent nodes are 18 cm and 16 cm respectively. (a) What is the length of the string? (b) If the tension is 10 N and the linear mass density is 4g/m, what is the fundamental frequency?

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Ans.

Q.11

Find the intensity of sound wave whose frequency is 250 Hz. The displacement amplitude of particles of the medium at this position is 1 × 10–8 m. The density of the medium is 1 kg/m3, bulk modulus of elasticity of the medium is 400 N/m2.

Ans.

Q.12

The loudness level at a distance R from a long linear source of sound is found to be 40dB. At this point, the amplitude of oscillations of air molecules is 0.01 cm. Then find the loudness level & amplitude at a point located at a distance '10R' from the source.

Ans.

Q.13

Sound of wavelength λ passes through a Quincke’s tube, which is adjusted to give a maximum intensity I0. Find the distance through the sliding tube should be moved to give an intensity I0/2.

Ans.

Q.14

The first overtone of a pipe closed at one end resonates with the third harmonic of a string fixed at its ends. The ratio of the speed of sound to the speed of transverse wave travelling on the string is 2 :1. Find the ratio of the length of pipe to the length of string.

Ans.

Q.15

A steel rod having a length of 1 m is fastened at its middle. Assuming young’s modulus to be 2 × 1011 Pa, and density to be 8 gm/cm3 find the fundamental frequency of the longitudinal vibration and frequency of first overtone.

Ans.

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Q.16

In a resonance-column experiment, a long tube, open at the top, is clamped vertically. By a separate device, water level inside the tube can be moved up or down. The section of the tube from the open end to the water level act as a closed organ pipe. A vibrating tuning fork is held above the open end, first and the second resonances occur when the water level is 24.1 cm and 74.1 cm repsectively below the open end. Find the diameter of the tube.

Ans.

Q.17

An open organ pipe filled with air has a fundamental frequency 500Hz. The first harmonic of another organ pipe closed at one end and filled with carbon dioxide has the same frequency as that of the first harmonic of the open organ pipe. Calculate the length of each pipe. Assume that the velocity of sound in air and in carbondioxide to be 330 and 264 m/s respectively.

Ans.

Q.18 Tuning fork A when sounded with a tuning fork B of frequency 480 Hz gives 5 beats per second. When the prongs of A are loaded with wax, it gives 3 beats per second. Find the original frequency of A. Ans.

Q.19

A car is moving towards a huge wall with a speed = c/10 , where c = speed of sound in still air. A wind is also blowing parallel to the velocity of the car in the same direction and with the same speed. If the car sounds a horn of frequency f, then what is the frequency of the reflected sound of the horn heared by driver of the car?

Ans.

Q.20

A fixed source of sound emitting a certain frequency appears as fa when the observer is approaching the source with speed v and frequency fr when the observer recedes from the source with the same speed. Find the frequency of the source.

Ans.

Q.21

Two stationary sources A and B are sounding notes of frequency 680 Hz. An observer moves from A to B with a constant velocity u. If the speed of sound is 340 ms–1, what must be the value of u so that he hears 10 beats per second?

Ans.

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SUBJECTIVE–II Q.1

(i) (ii) (iii) Ans.

Q.2 (a) (b) (c)

A long string under tension of 100 N has one end at x = 0. A sinusoidal wave is generated at x = 0 whose equation is given by  πx  y = (0.01 cm) sin   m  − 50 π t (sec)    10  1 Sketch the shape of the string at t = sec. 50 Find the average power transmitted by the wave. Draw velocity time graph of particle at x = 5 m.

The figure shows a snap photograph of a vibrating string at t = 0. The particle P is observed moving up with velocity 20 π cm/s. The angle made by string with x-axis at P is 6°. Find the direction in which the wave is moving the equation of the wave the total energy carried by the wave per cycle of the string , assuming that µ, the mass per unit length of the string = 50 gm/m.

Ans.

Q.3

A symmetrical triangular pulse of maximum height 0.4 m and total length 1 m is moving in the positive x-direction on a string on which the wave speed is 24 m/s. At t = 0 the pulse is entirely located between x = 0 and x = 1 m. Draw a graph of the transverse velocity of particle of string versus time at x =+1m.

Ans.

Q.4

A string fixed at both ends is vibrating in the lowest mode of vibration for which a point at quarter of its length from one end is a point of maximum displacement. The frequency of vibration in this mode is 100 Hz. What will be the frequency emitted when it vibrates in the next mode such that this point is again a point of maximum displacement?

Ans.

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Q.5

A steel of wire of length 25 cm is fixed at its ends to rigid walls. Young’s modulus of steel = 200 GPa, coefficient of linear thermal expansion = 10–5 / °C. Initially, the wire is just taut at 20°C temperature. The density of steel = 8.0 g/cc. A tuning fork of frequency 200 Hz is touched to the wire, to excite oscillations. Simultaneously, the temperature is slowly lowered. At what temperature will resonance occur corresponding to the third overtone?

Ans.

Q.6

A uniform rope of length L and mass m is held at one end and whirled in a horizontal circle with angular velocity ω. Ignore gravity. Find the time required for a transverse wave to travel from one end of the rope to the other.

Ans.

Q.7 (a) (b) (c)

The displacement of the medium in a sound wave is given by the equation ; y1 = A cos (ax + bt) where A, a & b are positive constants. The wave is reflected by an obstacle situated at x = 0. The intensity of the reflected wave is 0.64 times that of the incident wave. what are the wavelength & frequency of the incident wave. write the equation for the reflected wave. in the resultant wave formed after reflection , find the maximum & minimum values of the particle speeds in the medium.

Ans.

Q.8

A metal rod of length l = 100 cm is clamped at two points. Distance of each clamp from nearer end is a=30cm. If density and Young’s modulus of elasticity of rod material are ρ = 9000 kg m-3 and Y = 144 GPa respectively, calculate minimum and next higher frequency of natural longitudinal oscillations of the rod.

Ans.

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Q.9

(a) (b)

Two speakers are driven by the same oscillator with frequency of 200 Hz. They are located 4 m apart on a vertical pole. A man walks straight towards the lower speaker in a direction perpendicular to the pole, as shown in figure. How many times will he hear a minimum in sound intensity, and how far is he from the pole at these moments? Take the speed of sound to be 330 m/s, and ignore any sound reflections coming off the ground.

Ans.

Q.10 (a) (b) (c) (d) Ans.

Q.11

A source emits sound waves of frequency 1000 Hz. The source moves to the right with a speed of 32 m/s relative to ground. On the right a reflecting surface moves towards left with a speed of 64 m/s relative to the ground. The speed of sound in air is 332 m/s. Find the wavelength of sound in air incident on reflecting surface the number of waves arriving per second which meet the reflecting surface. the speed of reflected waves. the wavelength of reflected waves.

A train of length l is moving with a constant speed v along a circular track of radius R, The engine of the train emits a whistle of frequency f. Find the frequency heard by a guard at the rear end of the train. Make suitable assumption.

Ans.

Q.12

In a stationary wave pattern that forms as a result of reflection of waves from an obstacle the ratio of the amplitude at an antinode and a node is β=1.5. What percentage of the energy passes across the obstacle?

Ans.

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ANSWER KEY OBJECTIVE QUESTION BANK ONLY ONE OPTION IS CORRECT. Q.1

A

Q.2

A

Q.3

D

Q.4

C

Q.5

C

Q.6

B

Q.7

D

Q.8

A

Q.9

A

Q.10

D

Q.11

B

Q.12

B

Q.13

A

Q.14

B

Q.15

A

Q.16

C

Q.17

D

Q.18

B

Q.19

C

Q.20

D

Q.21

B

Q.22

C

Q.23

B

Q.24

A

Q.25

A

Q.26

B

Q.27

B

Q.28

A

Q.29 B

Q.30

A

Q.31

B

Q.32

C

Q.33

C

Q.34

D

Q.35

D

Q.36

A

Q.37

C

Q.38

B

Q.39 D

Q.40

C

Q.41

A

Q.42

B

Q.43

A

Q.44

C

Q.45

B

Q.46

D

Q.47

C

Q.48

C

Q.49

A

Q.50 A

Q.51

A

Q.52

B

Q.53

A

Q.54

D

Q.55

D

A

Q.6

A

ASSERTION AND REASON Q.1

D

Q.2

A

Q.3

D

Q.4

A

Q.5

ONE OR MORE THAN ONE OPTION MAY BE CORRECT Q.1

C

Q.2

B

Q.3

A, B, D

Q.4

C, D

Q.5

B, D

Q.6

B, C

Q.7

A, B

Q.8

A, C

Q.9

D

Q.10

C, D

Q.11

B

Q.12

A, C

Q.13

A, B, C, D

Q.14

B, C

Q.15

A, B, C, D

Q.16

A, C, D

Q.17

A, B

Q.18

C

Q.19

A

Q.20

A, B, D

Q.21

A, D

Q.22

C

Q.23

B, C

Q.24

C

Q.25

C

Q.26

A, B, C

Q.27

A, C, D

Q.28

B

Q.29

A

Q.30

B

Q.31

C, D

Q.32

C

Q.33

B, D

Q.34

B

Q.35

C

Q.36

C

Q.37

C

Q.38

B, C

Q.39

B, D

Q.40

B, D

Q.41

(A)–P,Q (B)–R,S (C)–P,S (D)–Q,R

ETOOS Academy Ltd. : F-106, Road no.2, Indraprastha Industrial Area, End of Evergreen Motors (Mahindra Showroom), BSNL Office Lane, Jhalawar Road, Kota, Rajasthan (324005) [38]

SUBJECTIVE–I Q.1

(a) DEF, (b) ABH, (c) CG, (D) AE

Q.2

1 3π 3π 2 mm/s, mm/s2 , (c) zero (a) λ = 4m, f = Hz, 1 m/s, (b) 4 2 4

Q.4

10.8 ×104W

Q.5

Ar = –

Q.7

1 × 109 Nm2

Q.8

50Hz, 50π cm/sec.

Q.10

(a) 144 cm ; (b) 17.36 Hz

Q.11

Q.13

λ/8

Q.14

Q.16

3 cm

Q.17

Q.19

11f / 9

Q.20

Q.3

1.22 v

Q.6

0.2 cm

Q.9

1%

Q.12

30 dB, 10 10 µm

1:1

Q.15

2.5 kHz, 7.5 kHz

33 cm and 13.2 cm

Q.18 485 Hz

1 2 cm, At = cm 3 3

π 2 × 10 −9 W/m2 4

f r + fa

Q.21

2

2.5 ms–1

SUBJECTIVE–II (ii) 25 × 10–6 W

Q.1

(i)

Q.2

1 (a) negative x ; (b) y = 4×10–3 sin 100π  3t + 0.5x + 400 

Q.3

Q.6

Q.4

π 2ω

(iii)

, solved

 (x , y in meter) ; (c) 144π2 ×10–5 J  

300 Hz

Q.5

17.5°

Q.7 (a) 2 π/a, b/2π, (b) y2 = ± 0.8 A cos (ax - bt), (c) max.=1.8 b A, min. = 0,

Q.8

10kHz, 30kHz

Q.9

(a) 2 ; (b) 9.28 m and 1.99 m

Q.10

(a) 0.3 m, (b) 1320, (c) 332 m/s, (d) 0.2 m

Q.11

f

Q.12

96 %

ETOOS Academy Ltd. : F-106, Road no.2, Indraprastha Industrial Area, End of Evergreen Motors (Mahindra Showroom), BSNL Office Lane, Jhalawar Road, Kota, Rajasthan (324005) [39]