Mechanical Waves | Visual Physics for IIT JEE / NEET
Introduction

What is a Wave ? How is a Mechanical Wave different from an ElectroMagnetic Wave ? What do we mean by Transverse and Longitudinal Waves ? Do waves transport Energy is apace without transporting matter ?

Sinusoidal Wave Eq

Visualizing a Transverse Sinusoidal Wave as a projection of particles circulating a little out of phase. Introduction to Wavelength, Angular Wave number, Frequency. Equation of a Traveling Wave.

Q1

A wave is described by y = ( 50 cm ) sin [ ( 2 rad / m) x - ( 4 rad / sec ) t ]. Determine the amplitude, wavelength, frequency, time period, speed and direction of the wave ?

Q2

The equation of a wave travelling in a string can be written as y = 3 cos p (100t - 2x). Its wavelegth is

Q3

A travelling sinusoidal wave is generated by an oscillator that completes 50 vibrations in 10 s. Distance travelled by the wave in 10 sec is = 4m. What is the wavelength of the wave?

General Wave Eq

General Wave Equation.

Q4

A pulse is travelling in + ve direction with a speed of 2m/s. Displacement y of the particle at x = 0 at any time t is given by Find i) expression of the function y = (x, t), i.e., displacement of a particle at position x and time t. ii) shape of the pulse at t = 0 and t = 6 s.

Q5

At t = 0, transverse pules in a wire is described by the function Where x and y are in meters. Write the function y(x, t) that describes the pulse if it is travelling with a speed of 2 m/s in a) positive x-direction b) negative x-direction

Diff Wave Eq

Differential Wave Equation

Q6

Verify that the given equation is a wave function.

Phase Difference

Phase difference between two waves

Speed on String

Derivation and Understanding of the Equation of Speed of a wave on a string.

Q7

Transverse waves travel with a speed of 10 m/s in a string under a tension of 50 N. What tension is required for a wave speed of 30 m/s in the same string?

Q8

Two strings A and B, made of same material, are stretched by same tension. The radius of string A is double the radius of B. A transverse wave travels on A with speed vA and on B with speed vB. The ratio vA /vB is

Superposition

Principle of Superposition of Waves. How do waves affect each other ?

Interference

Derivation of Equation of Interference of waves.

Standing Waves

What are Standing waves ? And can a wave be standing ?

Q9

The equation of a stationary wave produced on a string whose both ends are fixed is given by y = [ (0.6 cm) sin ( p /10 cm-1 ) x ] cos (600 p s-1) t What could be the smallest length of the string?

Q10

In a stationery wave system, all the particles of the medium (a) have zero displacement simultaneously at some instant (b) have maximum displacement simultaneously at some instant (c) are at rest simultaneously at some instant (d) reach maximum velocity simultaneously at some instant (e) vibrate in the same phase (f) in the region between two antinodes vibrate in the same phase (g) in the region between two antinodes vibrate in the same phase (h) on either side of a node vibrate in opposite phase

Q11

A string of length L is stretched along the x axis and is rigidly clamped at its two ends. It undergoes transverse vibration. If n is an integer, which of the following relations may represent the shape of the string at any time t?

Q12

Show that two superimposed waves of the same frequency and amplitude traveling in the same direction cannot give rise to a standing wave.

Reflection

A wave can reflected either inverted or non-inverted, depending on whether the end is free to move or not.

Q13

A string of length l and linear mass density m is clamped at its ends. The tension in the string is T. When a pulse travel along the string, the shape of string is found to be the same at times t + Dt. Find the value of Dt

Q14

Consider the following wave function: a) y = A sin ( wt - kx ) b) y = A sin ( kx - wt ) c) y = A cos ( wt - kx ) d) y = A cos ( kx - wt ) e) y = A sin ( kx + wt ) f) y = A cos ( kx + wt ) Write the equations of reflected wave after reflection from a free and a fixed boundary. Also find the resulting stationary wave formed by the superposition of its reflected wave.

Modes of Vibration

Modes or Harmonics or Vibrations of a Standing wave.

Q15

A string has length of 10 m and a mass of 25 gm. If the tension in the string is 100 N, what are the three lowest frequencies it produces when plucked?

Sonometer

Functioning of a Sonometer

Q16

A sonometer wire of length l vibrates in fundamental mode when exited by tuning fork of frequency 416 Hz. If the length is doubled keeping other things same the string will

Transmission

Transmission of a Wave from one medium to other. Properties of the Reflected and Transmitted waves.

Q17

A long wire PQR is made by joining two wires PQ and QR of equal radii. PQ has length 4.8 m and mass 0.06 kg. QR has length 2.56 m and mass 0.2 kg. The wire PQR is under a tension of 80 N. A sinusoidal wave-pulse of amplitude 3.5 cm is sent along the wire PQ from the end P. Calcutate the time taken by the wave-pulse to reach the other end R of the wire.

Energy Power - 1

Propagation of Kinetic and Potential Energy in a Wave. Concept of Energy Density.

Energy Power - 2

Energy Power and Intensity of a wave

Q18

A string with linear mass density m = 400 x 10-2 kg/m is under a tension of 100 N. How much power must be supplied to the string to generate sinusoidal waves at a frequency of Hz and an amplitude of 500 cm ? What is the energy stored in 2 m length of wire ?

Q19

Two waves travelling in the same medium are represented by y-t graphs in the figure. Find ratio of their average intensities?

Q20

Mark out the correct options. a) The energy of any small part of a string remains constant in a travelling wave b) The energy of any small part of a string remains constant in a standing wave c) The energies of all the small part of equal length are equal in a traveling wave d) the energies of all the small parts of equal length are equal in a standing wave

1

A transverse sinusoidal wave moves along a string in the positive x-direction at a speed of 10 cm s-1. The wavelength of the wave is 0.5 m and its amplitude is 10 cm. At a particular time t, the snap-shot of wave is shown in figure. The velocity of point P when its displacement is 5 cm is

2

Graph shows the snapshot of a sinusoidal travelling wave. It is known that the wave is travelling in +ve x direction and has frequency = 10 Hz. a) Determine the amplitude, wavelength, angular wave number, angular frequency and speed of the wave. b) Write the general equation of wave. c) What is the maximum transverse speed and acceleration of a particle on wave ?

3

(a)Write the expression for y as a function of x and t for a sinusoidal wave travelling along a rope in the negative x direction with the following characteristics: A = 10 cm, l = 1 m , f = 3 Hz, and y(0, t) = 0 at t = 0. (a) assuming that y(x, 0) = 0 at the point x = 10 cm (b) Write an expression for y as a function of x and t for the wave in part

4

Figure shows the shape of a progressive wave at time t = 0. After a time t = 0.5 s, the particle at the origin has its maximum negative displacement. If the wave speed is 4 m/s, then find the equation of the progressive wave.

5

The wave function for a travelling wave on a taut string is y(x,t) = (2m) sin(5?t ? 2?t + ?/4). ( SI units) (a) what is the direction of travel of the wave? (b) what are the amplitude, wave length, frequency and speed of the wave? (c) what is maximum transverse speed and acceleration of an element of the string? (d) plot the wave at t = 0. (e) what is vertical position of an element of the string at t = 0, x = 0.5 m?

6

A, B, C are the three particles equally separated and lie along the x-axis. When a sinusoidal transverse wave of wavelength l propagates along the x-axis, the following observation are made: A and C have the same velocity. A and B have the same speed. Find : (i) The minimum distance between A and B (ii) The minimum distance between A and C.

7

A progressive wave of frequency 500 Hz is travelling with a velocity of 300 m/s. a) how far apart are two points who have an absolute phase difference of 60o ? effective phase difference of 60o ? b) what is the phase difference between two points which are separated by distance of 20 cm along the direction of wave propagation. c) by how much does the phase of a particle change in 4 ms

8

A sinusoidal wave is propagating along a stretched string that lies along the x-axis. The displacement of the string as a function of time is graphed in figure for particles at x = 0 and at x = 10 cm. ( Given l > 10 cm ) (a) what is amplitude of the wave? (b) what is the period of the wave? (c) determine the wave length and wave speed, if the wave is moving in i) +x-direction ii) ?x-direction

9

Figure shown two snapshots of medium particles within a time interval of 1/50 s. Find the possible time periods of the wave

10

A sinusoidal wave traveling in the positive direction on a stretched string has amplitude 2.0 cm, wavelength 1.0 m, and wave velocity 5.0 m/s. At x = 0 and t = 0, it is given that y = 0 and < 0. Find the wave function y = f (x, t).

11

A string A has double the length, double tension, double the diameter and double the density as another string B. The ratio of their fundamental frequencies of vibration is equal to

12

A metallic wire is clamped at each end under zero tension initially. What strain will result in transverse wave speed of v m/s? Given the cross-sectional area of the wire to be A m the density to be m kg/m3, and Young?s modulus to be Y N/m3.

13

The length of a sonometer wire between two fixed ends is 2.2 m. Where should the two bridges be placed to divide the wire into three segments whose fundamental frequencies are in the ration of 1:2:3?

14

The following equations represent transverse waves: Z1 = A cos ( k x - w t ) Z2 = A cos ( k x + w t ) Z3 = A cos ( k y - w t ) Identify the combination of the waves which will produce a) standing wave (s) b) a wave travelling in the direction making an angle of 45o with positive y-axis. In each case, find the position at which the resultant wave intensity is always zero.

15

A wave represented by the equation y = a cos ( kx ? wt ) is superposed with another wave to form a stationary wave such that the point x = 0 is a node. The equation of other wave is

16

A stone of density s hangs from the free and of a sonometer wire. The fundamental frequency of vibration of wire is f1,a. If the stone hangs wholly immersed in a liquid of density r, the fundamental frequency become

17

The extension in a string, obeying Hooke?s law, is x. The speed of the wave in the stretched string is v. If the extension in the string is increased to 1.5 x, the speed of the wave in the string will be (a) 1.22 v (b) 0.61 v (c) 1.50 v (d) 0.75 v

18

Two wires having identical geometrical construction, are stretched from their natural length by small but equal amount. The young modulus of the wires are Y1 and Y2 whereas the densities are r1 and r2. It is given that Y1 > Y2 and r1 > r2. A transverse signal started at one end takes a time t1 to reach the other end for 1st wire and t2 for 2nd wire. Then a) t1 < t2 b) t1 = t2 c) t1 > t2 d) the information is insufficient to find the relation between t1 and t2

19

A sonometer wire resonates with a given tuning fork forming standing waves with five antinodes between the two bridges when a mass of 9 kg is suspended from the wire. When this mass is replaced by a mass m, the wire resonates with the same tuning fork forming three antinodes for the same positions of the bridges. The value of m is a) 25 kg b) 5 kg c) 12.5 kg d) (1/25) kg

20

Two pulses in a stretched string whose centers are initially 8 cm apart are moving towards each other as shown the figure. The speed of each pulse is 2 cm/s. After two seconds the total energy of the pulses will be

21

The end of a stretched wire of length L are fixed at x = 0 and x = L. In one experiment, the displacement of the wire is y1 = A sin ( p x / L ) sin ( w t ) and energy is E1. In another experiment, the displacement is y2 = A sin ( 2 p x / L ) sin ( 2 w f ) and energy is E2. Then a) E2 = E1 b) E2 = 2 E1 c) E2 = 4E1 d) E2 = 16 E1

22

A uniform string ( length L, linear density m, and tension T ) is vibrating with amplitude A in its nth mode. Shown that its total energy oscillation is given by

23

A wave given by Yi = A sin (wt ? (wx)/v)is sent down a string. Upon reflection it becomes Yr = ? A sin (wt + (wx)/v). Show that the resultant of two waves on the string can be written as combination of standing wave and traveling wave.

24

Two metallic strings A and B of different materials are connected in series forming a joint. The strings have similar cross-sectional area. The length of A is lA 0.3 m and that of B is lB = 0.8 m. Other ends of the wires are fixed. Transverse wave is set up in the combined string using an external source of variable frequency. For a standing wave to occur in the composite string, find

25

Sources separated by 20 m vibrate according to the equations y1 = 1/2 sin ( ?t ) m y2 = 1/4 sin ( ?t ) m The send out waves travel at the speed of 3 m/s. What is the equation of motion of a particle 12 m from the first source ?

26

The vibration of a spring of length 60 cm fixed at both ends is represented by the equation Y = 4 sin(?x/15 cos(96 ?t) (a) where are the nodes located along the string ? (b) what is maximum displacement of a point x = 5 cm ? (c) what is the velocity of particle at x = 7.5 cm at t = 0.25 sec ? (d) write the down the equation of the component waves whose superposition gives the above wave.

27

A rope of total mass m and length L is suspended vertically. Find the time taken to travel a) entire length of rope b) lower half of rope c) upper half of rope d) What is the distance travelled by pulse in half of the time in part (a)

28

A wave pulse starts propagating in the +x - direction along a non - uniform wire of length 10 m with mass per unit length given by m = mo + ax and under a tension of 100 N. Find the time taken by a pulse to travel from the lighter end ( x = 0 ) to the heavier end. ( mo = 5 10-2 kg/m and a = 10-2 kg/m2)

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