Waves

Let mass per unit length of wires are $\mu $₁ and $\mu $₂ respectively

$ \because $ Materials are same, so density $\rho $ will be same.

$ \therefore $ ${\mu _1} = {{\rho \pi {r^2}L} \over L} = \mu $ and ${\mu _2} = {{\rho 4\pi {r^2}L} \over L} = 4\mu $

Tension in both are same = T, let speed of wave in wires are V₁ and V₂

${V_1} = {{{V_1}} \over {2L}} = {V \over {2L}}\,\,\& \,{V_2} = {{{V_2}} \over {2L}} = {V \over {4L}}$

Frequency at which both resonate is L.C.M. of both frequencies (i.e : ${V \over {2L}}$ )

Hence number of loops in wires are 1 and 2 respectively

So, ratio of number of antinodes is 1 : 2.

In first resonance, length of air column $ = {\lambda \over 4}$.

So, ${l_1} + e = {\lambda \over 4}$ or $11 \times 4 + 4e = \lambda $

So, speed of sound is

$ \Rightarrow v = {f_1}\lambda = 512(44 + 4e)$ ...... (i)

And in second case,

$l{'_1} + e = {{\lambda '} \over 4}$ or $27 \times 4 + 4e = \lambda '$

$ \Rightarrow v = {f_2}\lambda ' = 256(108 + 4e)$ ..... (ii)

Dividing both Eqs. (i) and (ii), we get

$1 = {{512(44 + 4e)} \over {256(108 + 4e)}} \Rightarrow e = 5$ cm

Substituting value of e in Eq. (i), we get

Speed of sound $v = 512(44 + 4e)$

$ = 512(44 + 4 \times 5)$

$ = 512 \times 64$ cm s^$-$1 = 327.68 ms^$-$1 $\approx$ 328 ms^$-$1

y = a sin($\omega $t + kx)

$ \Rightarrow $  wave is moving along $-$ve x-axis with speed

v = ${\omega \over K}$ $ \Rightarrow $  v = ${{50} \over 2}$ = 25m/sec

y = 0.03 sin(450 t $-$ 9x)

v = ${\omega \over k} = {{450} \over 9}$ = 50m/s

v = $\sqrt {{T \over \mu }} \Rightarrow {T \over \mu }$ = 2500

$ \Rightarrow $  T = 2500 $ \times $ 5 $ \times $ 10^$-$3

= 12.5 N

Velocity of wave on string

$V = \sqrt {{T \over \mu }} = \sqrt {{8 \over 5} \times 1000} = 40m/s$

Now, wavelength of wave

$\lambda = {v \over n} = {{40} \over {100}}m$

Separation b/w successive nodes,

${\lambda \over 2} = {{20} \over {100}}\,m$ $=$ 20 cm

f_app = f₀ $\left[ {{{{v_2} \pm {v_0}} \over {{v_2} \pm {v_s}}}} \right]$

f₁ = f₀ $\left[ {{{340} \over {340 - 34}}} \right]$

f₂ = f₀ $\left[ {{{340} \over {340 - 17}}} \right]$

${{{f_1}} \over {{f_2}}} = {{340 - 17} \over {340 - 34}} = {{323} \over {306}} \Rightarrow {{{f_1}} \over {{f_2}}} = {{19} \over {18}}$

Frequency of sound wave produce by flute

= ${{2{V_S}} \over {2\ell }}$

= ${{2 \times 330} \over {2 \times 50 \times {{10}^{ - 2}}}}$

= 660 Hz

Speed of the observer

= 10km/hr

= 10 $ \times $ ${5 \over {18}}$ m/s

= ${{25} \over 9}$ m/s

Frequency heard by the observer,

f' = $\left( {{{{V_S} + {V_o}} \over {{V_S}}}} \right)f$

= $\left( {{{330 + {{25} \over 9}} \over {330}}} \right) \times 660$

= 666 Hz

Resultant force on the ball of mass M when car is moving with a acceleration a is ,

F_net = $\sqrt {{{\left( {Mg} \right)}^2} + {{\left( {Ma} \right)}^2}} $

   = $M\sqrt {{g^2} + {a^2}} $

$ \therefore $   T = M$\sqrt {{g^2} + {a^2}} $

We know,

Velocity, V = $\sqrt {{T \over \mu }} $

When Car is at rest then,

   60 = $\sqrt {{{Mg} \over \mu }} $   . . . . (1)

and when is moving then

   60.5 = $\sqrt {{{M\sqrt {{g^2} + {a^2}} } \over \mu }} $    . . . . (2)

By dividing (2) by (1) we get,

${{60.5} \over {60}} = \sqrt {{{\sqrt {{g^2} + {a^2}} } \over g}} $

$ \Rightarrow $   $\left( {1 + {{0.5} \over {60}}} \right)$ = ${\left( {{{{g^2} + {a^2}} \over {{g^2}}}} \right)^{{1 \over 4}}}$

$ \Rightarrow $   ${{{g^2} + {a^2}} \over {{g^2}}}$ = ${\left( {1 + {{0.5} \over {60}}} \right)^4}$

$ \Rightarrow $   ${{{g^2} + {a^2}} \over {{g^2}}}$ = 1 + 4 $ \times $ ${{{0.5} \over {60}}}$ [Using Binomial approximation]

$ \Rightarrow $   ${{{g^2} + {a^2}} \over {{g^2}}}$ = 1 + ${1 \over {30}}$

$ \Rightarrow $   1 + ${{{a^2}} \over {{g^2}}}$ = 1 + ${1 \over {30}}$

$ \Rightarrow $   ${a \over g}$ = ${1 \over {\sqrt {30} }}$

$ \Rightarrow $   a = ${g \over {\sqrt {30} }}$

$ \therefore $   Closest answer, a = ${g \over 5}$

Fundamental frequency is maximum when length is minimum i.e. L₀,

Case 1. $L = {L_0},\,T = {T_0},\,f = {f_0}$;

${f_1} = {1 \over {2{L_0}}}\sqrt {{{{T_0}} \over \mu }} $

Case 2. ${f_2} = {1 \over {{L_0}}}\sqrt {{{{T_2}} \over {2\mu }}} = {{{f_0}} \over {\sqrt 2 }}$

Case 3. ${f_3} = {1 \over {{L_0}}}\sqrt {{{{T_2}} \over {3\mu }}} = {{{f_0}} \over {\sqrt 3 }}$

Case 4. ${f_4} = {1 \over {{L_0}}}\sqrt {{{{T_2}} \over {4\mu }}} = {{{f_0}} \over 2}$

Case 1. $L = {L_0},\,T = {T_0},\,f = {f_0}$

${f_1} = {1 \over {2{L_0}}}\sqrt {{{{T_0}} \over \mu }} $

Case 2. $L = {{3{L_0}} \over 2}$

${f_2} = {3 \over {2 \times {{3{L_0}} \over 2}}}\sqrt {{{{T_2}} \over {2\mu }}} = {f_0}$

$ \Rightarrow {f_0} = {1 \over {2{L_0}}}\sqrt {{{{T_2}} \over \mu }} \Rightarrow {T_2} = {{{T_0}} \over 2}$

Case 3. $L = {{5{L_0}} \over 4}$

${f_3} = {5 \over {2 \times {{5{L_0}} \over 4}}}\sqrt {{{{T_3}} \over {3\mu }}} = {f_0}$

$ \Rightarrow {f_0} = {2 \over {\sqrt 3 {L_0}}}\sqrt {{{{T_3}} \over \mu }} \Rightarrow {T_3} = {{3{T_0}} \over {16}}$

Case 4. $L = {{7{L_0}} \over 4}$

${f_4} = {{14} \over {2 \times {{7{L_0}} \over 4}}}\sqrt {{{{T_4}} \over {4\mu }}} = {f_0}$

$ \Rightarrow {f_0} = {2 \over {{L_0}}}\sqrt {{{{T_4}} \over \mu }} \Rightarrow {T_4} = {{{T_0}} \over {16}}$

Given, End correction (e) = 1 cm

For first resonance,

${\lambda \over 4} = {l_1} + e$ = 10 + 1 = 11 cm

For second resonance,

${3\lambda \over 4} = {l_2} + e$

$ \Rightarrow $ ${l_2}$ = 3 $ \times $ 11 - 1 = 32 cm

Frequency of B, f_B = 425 $ \pm $ 5 = 420 or 430 Hz

As tension of string B is increased

So, frequency of B, f_B should also increase [as f $ \propto $ $\sqrt T $]

If initially f_B = 430 Hz then when f_B increases by increasing the tension then f_B $-$ f_A increases that means beat frequency increase.

So, f_B can't be 430 Hz

When f_B = 420 then when f_B increases f_A $-$ f_B decreases means beat frequency decreases. So, correct f_B = 420 Hz.

As   rod length = 60 cm

$\therefore\,\,\,$ ${\lambda \over 2}$ = 60

$ \Rightarrow $$\,\,\,$ $\lambda $ = 120 cm = 1.2 m

In solid, velocity of wave,

V = $\sqrt {{Y \over \rho }} $

= $\sqrt {{{9.27 \times {{10}^{10}}} \over {2.7 \times {{10}^3}}}} $

= 5.85 $ \times $ 10³ m/sec.

As  we know,

v = f $\lambda $

$\therefore\,\,\,$ f = ${v \over \lambda }$

= ${{5.85 \times {{10}^3}} \over {1.2}}$

= 4.88 $ \times $ 10³ Hz

$ \simeq $  5 kHz

Length of wire is L₁ = 0.95 m; L₂ = 1 m. Number of beats per second heard = 5

Let frequency of fork be f. Therefore,

${v \over {2{L_1}}} - f = 5 \Rightarrow {v \over {2{L_1}}} = 5 + f$ ...... (1)

and $f - {v \over {2{L_2}}} = 5 \Rightarrow {v \over {2{L_2}}} = f - 5$ .... (2)

Dividing Eq. (1) by Eq. (2), we get

${{{v \over {2{L_1}}}} \over {{v \over {2{L_2}}}}} = {{5 + f} \over {f - 5}} \Rightarrow {{{L_2}} \over {{L_1}}} = {{f + 5} \over {f - 5}}$

$ \Rightarrow {1 \over {0.95}} = {{f + 5} \over {f - 5}} \Rightarrow f - 5 = 0.95f + 4.75$

$ \Rightarrow f - 0.95f = 5 + 4.75 \Rightarrow 0.05f = 9.75 \Rightarrow f = {{9.75} \over {0.05}}$

$ \Rightarrow f = 195$ Hz

The tuning fork vibrates with frequency 256 Hz and give one beat per second So, the organ pipe will have frequency (256 $ \pm $ 1) Hr.

For open organ pipe,

Frequency n = ${{N\upsilon } \over {2\ell }}$

Here n = 255 Hz

N = 3

$\upsilon $ = 340 m/s

$\therefore\,\,\,\,$ 255 = ${{3 \times 340} \over {2 \times \ell }}$

$ \Rightarrow $$\,\,\,\,$ $\ell $ = ${{3 \times 340} \over {2 \times 255}} = 2\,m$

$\therefore\,\,\,\,$ $\ell $ = 2m or 200 cm

To determine which statements are true, we'll analyze the experiment step by step, using the theory of resonating air columns in a closed tube (closed at one end and open at the other).

Fundamentals of Resonance in a Closed Tube

For a tube closed at one end, the resonant lengths occur at:

$ L_n = \left( \frac{2n - 1}{4} \right) \lambda + e $

where:

$ L_n $ is the length of the air column for the $ n^\text{th} $ resonance,

$ \lambda $ is the wavelength of the sound wave,

$ e $ is the end correction,

$ n = 1, 2, 3, \dots $

Successive resonances (like those at $ L_1 $ and $ L_2 $) can be used to eliminate the end correction and find the wavelength.

Calculations

Step 1: Calculate the Wavelength ($ \lambda $)

Given:

First resonance length ($ L_1 $) = 50.7 cm

Second resonance length ($ L_2 $) = 83.9 cm

The difference between successive resonances in a closed tube is:

$ L_2 - L_1 = \frac{\lambda}{2} $

Solving for $ \lambda $:

$ \lambda = 2(L_2 - L_1) = 2(83.9\, \text{cm} - 50.7\, \text{cm}) = 2(33.2\, \text{cm}) = 66.4\, \text{cm} $

Conclusion: The wavelength of the sound wave is 66.4 cm.

Option C is true.

Step 2: Calculate the Speed of Sound ($ v $)

Using the relationship $ v = f\lambda $:

Frequency ($ f $) = 500 Hz

Wavelength ($ \lambda $) = 66.4 cm = 0.664 m

$ v = f\lambda = 500\, \text{Hz} \times 0.664\, \text{m} = 332\, \text{m/s} $

Conclusion: The speed of sound determined from the experiment is 332 m/s.

Option A is true.

Step 3: Calculate the End Correction ($ e $)

Assuming the first resonance corresponds to the second harmonic ($ n = 2 $):

$ L_1 = \left( \frac{2n - 1}{4} \right) \lambda + e $

For $ n = 2 $:

$ L_1 = \left( \frac{3}{4} \right) \lambda + e $

Solving for $ e $:

$ e = L_1 - \left( \frac{3}{4} \right) \lambda = 50.7\, \text{cm} - \left( \frac{3}{4} \times 66.4\, \text{cm} \right) $

$ e = 50.7\, \text{cm} - 49.8\, \text{cm} = 0.9\, \text{cm} $

Conclusion: The end correction is 0.9 cm.

Option B is true.

Step 4: Determine if $ L_1 $ is the Fundamental Harmonic

Assuming $ L_1 $ corresponds to the fundamental harmonic ($ n = 1 $):

$ L_1 = \left( \frac{1}{4} \right) \lambda + e $

$ e = L_1 - \left( \frac{1}{4} \right) \lambda = 50.7\, \text{cm} - 16.6\, \text{cm} = 34.1\, \text{cm} $

An end correction of 34.1 cm is unreasonably large for typical experiments, indicating that $ L_1 $ does not correspond to the fundamental harmonic.

Conclusion: The resonance at 50.7 cm does not correspond to the fundamental harmonic.

Option D is false.

Standard equation of standing wave,

y(x, t) = 2a sin kx cos $\omega $t

Given,

y(x, t) = 0.5 sin $\left( {{{5\pi } \over 4}x} \right)$ cos (200$\pi $t).

So, k = ${{5\pi } \over 4}$ and $\omega $ = 200$\pi $

$\therefore\,\,\,$ Speed of travelling wave

= ${\omega \over k}$ = ${{200\pi } \over {{{5\pi } \over 4}}}$ = 160 m/s.

The time period is given by the formula

$T = 2\pi \sqrt {{l \over g}} $

which clearly indicates that the time period is directly proportional to the length of the pendulum l, that is,

$T \propto \sqrt l $

Here, l = 1 m. Therefore,

${{\Delta T} \over l} = {1 \over 2}{{\Delta l} \over l}$ ....... (1)

where $\Delta l = \left| {{r_1} - {r_2}} \right|$. Substituting the values in Eq. (1), we get

$5 \times {10^{ - 4}} = {1 \over 2}\left( {{{{r_1} - {r_2}} \over 1}} \right)$

$ \Rightarrow {r_1} - {r_2} = 10 \times {10^{ - 4}} = {10^{ - 3}}$ m $ = {10^{ - 1}}$ cm $ = 0.1$ cm

Fundamental frequency of a stretched string is ,

    f = ${n \over {2L}}\sqrt {{T \over \mu }} $

Here, n = number of antinodes

$\mu $ = mass per unit length.

As, O is the midpoint of two bridegs, hence length of two wires are equal.

$\therefore\,\,\,$ L₁ = L₂ = L

As frequency of both wires same

f₁ = f₂

$ \Rightarrow $$\,\,\,$ ${{{n_1}} \over {2L}}\sqrt {{T \over {\pi {r^2}{\rho _1}}}} $ = ${{{n_2}} \over {2L}}\sqrt {{T \over {\pi {r^2}{\rho _2}}}} $

$ \Rightarrow $$\,\,\,$ ${{{n_1}} \over {{n_2}}}$ = $\sqrt {{{{\rho _1}} \over {{\rho _2}}}} $

$ \Rightarrow $$\,\,\,$ ${{{n_1}} \over {{n_2}}}$ = $\sqrt {{{{\rho _1}} \over {4{\rho _1}}}} $ = ${1 \over 2}$

This question is from Doppler's effect of light.

When observer is moving towards the source then the frequency of wave measured by the observer will be

f_observed = f_actual$\sqrt {{{c + v} \over {c - v}}} $

where c = speed of light and v = speed of observer

According to the question, v = ${c \over 2}$

$\therefore$ f_observed = f_actual$\sqrt {{{c + {c \over 2}} \over {c - {c \over 2}}}} $

                     = 10$ \times $$\sqrt {{{{{3c} \over 2}} \over {{c \over 2}}}} $

                     = 10$\sqrt 3 $ = 17.3 GHz

The velocity of the transverse wave is given by

$v = \sqrt {{T \over \mu }} $ ..... (i)

Where T is the tension in the rope and is given by

T = Mg + $\mu$yg

Where y = distance of point P on the rope from its bottom

L = total length of the rope

Put the value of T in eqn. (i), we get

$v = \sqrt {{{Mg + \mu yg} \over \mu }} $ ...... (ii)

Also, speed of pulse 1 and pulse 2 are same at each point because speed is a characteristic of the medium. Hence, speed of the two pulses are same at the mid point (x = l/2). However, since pulse 1 is moving in +x direction and pulse 2 is moving in −x direction, their velocities (which is a vector quantity) at the mid point are not same.

So, option (b) is incorrect.

Since both the pulse travel in the same medium and velocity is medium dependent. The velocity of any pulse along the rope depends on tension in the rope and linear mass density so it is independent of its frequency(characteristic of the source of pulse generation) and wavelength.

So, option (d) is correct.

Also, v = $\nu $$\lambda$ or $\lambda$ $\propto$ v

($\because$ frequency of a wave depends on the source)

For pulse 1, the velocity decreases at point A as tension decreases. So the wavelength of pulse 1 becomes shorter when it reaches point A.

So, option (c) is incorrect.

From eqn. (ii), the velocity is given as

$v = \sqrt {{{Mg + \mu yg} \over \mu }} $ or ${{dy} \over {dt}} = \sqrt {{{Mg + \mu yg} \over \mu }} \Rightarrow dt = {{dy} \over {\sqrt {{{Mg + \mu yg} \over \mu }} }}$

Integrating both sides,

$\int\limits_0^T {dt = \int\limits_0^L {{{dy} \over {\sqrt {{{Mg} \over \mu } + yg} }}} } $; $\therefore$ $T = {2 \over g}\left( {\sqrt {{{Mg} \over \mu } + Lg} - \sqrt {{{Mg} \over \mu }} } \right)$

Where T is time taken by pulse to travel distance L.

Since T depends upon g, M, L and $\mu$ which are constant.

$\therefore$ Time taken by pulse 1 to reach point A and time taken by pulse 2 to reach point O is same.

$\therefore$ T_AO = T_OA; So option (a) is correct.

Let the frequency of the horn = f

Apparent frequency heared by the observer directly,

${f_{dir}} = \left( {{V \over {V - {V_s}}}} \right)f = \left( {{{340} \over {340 - 5}}} \right)f = {{340} \over {335}}f$

Apparent frequency heared by the observer on reflection from the wall,

${f_{ind}} = \left( {{V \over {V + {V_s}}}} \right)f = \left( {{{340} \over {340 + 5}}} \right)f = {{340} \over {345}}f$

Also, given that,

f_ind $-$ f_dir $=$ 5

$ \Rightarrow $   ${{340f} \over {345}}$ $-$ ${{340f} \over {335}}$ $=$ 5

$ \Rightarrow $   f $=$ ${5 \over {340}} \times {{335 \times 345} \over {10}}$

= 169.9  $ \simeq $   170 Hz

Frequency heard by the driver of second engine,

F' = $\left( {{{v + {v_0}} \over {v - {v_S}}}} \right)$ f

given v₀ = v_s = 30 m/s

$ \therefore $   f'  = $\left( {{{330 + 30} \over {330 - 30}}} \right) \times 540$

=   648 Hz

We know that velocity in string is given by

$v = \sqrt {{T \over \mu }} \,\,\,\,\,\,\,\,\,\,\,\,....\left( i \right)$

where $\mu = {m \over {\rm I}} = {{mass\,\,\,of\,\,\,string} \over {length\,\,\,of\,\,\,string}}$

The tension $T = {m \over \ell } \times x \times g\,\,\,\,\,\,\,\,\,\,...\left( {ii} \right)$

From $(a)$ and $(b)$ ${{dx} \over {dt}} = \sqrt {gx} $

${x^{ - 1/2}}\,dx = \sqrt g \,dt$

$\therefore$ $\int\limits_0^\ell {{x^{ - 1/2}}} dx - \sqrt g \int\limits_0^\ell {dt} $

$2\sqrt \ell = \sqrt g \times t$

$\therefore$ $t = 2\sqrt {{\ell \over g}} = 2\sqrt {{{20} \over {10}}} = 2\sqrt 2 $

The fundamental frequency in case $(a)$ is $f = {v \over {2\ell }}$

The fundamental frequency in case $(b)$ is

$f'{v \over {4\left( {\ell /2} \right)}} = {u \over {2\ell }} = f$

Consider the time $t\left(\leq t_Q\right)$ when car is at $S_1$ between $P$ and $Q$. The distance travelled by the car in time $t$ is $\mathrm{PS}_1=u t$. At this instant, the lines $\mathrm{S}_1{N}$ and $\mathrm{S}_1 \mathrm{M}$ both make angle $\theta$ with the velocity vector $\vec{u}$. The component of observer (person sitting in the car) velocity towards the sources $N$ and $M$ is $u_o=u \cos \theta$. The sources $N$ and $M$ are at rest i.e., $u_s=0$. Apply Doppler's effect equation to get frequencies of the sources $N$ and $M$ heard by the observer as

$ \begin{aligned} \nu_N^{\prime} & =\frac{v+u_o}{v-u_s} \nu_N=\frac{v+u \cos \theta}{v} \nu_N \\\\ & =\left(1+\frac{u}{v} \frac{D-u t}{\sqrt{d^2+(D-u t)^2}}\right) \nu_N, \\\\ \nu_M^{\prime} & =\frac{v+u_o}{v-u_s} \nu_M=\frac{v+u \cos \theta}{v} \nu_M \\\\ & =\left(1+\frac{u}{v} \frac{D-u t}{\sqrt{d^2+(D-u t)^2}}\right) \nu_M . \end{aligned} $

The beat frequency heard by the observer at time $t(\leq$ $\left.t_Q\right)$ is

$ \begin{aligned} \nu(t) & =\nu_N^{\prime}-\nu_M^{\prime} \\\\ & =\left(1+\frac{u}{v} \frac{D-u t}{\sqrt{d^2+(D-u t)^2}}\right)\left(\nu_N-\nu_M\right) ........(1) \end{aligned} $

Now, consider the time $t\left(\geq t_Q\right)$ when car is at $S_2$ between $Q$ and $R$. The distance travelled by the car in time $t$ is $\mathrm{PS}_2=u t$. At this instant, the lines $\mathrm{S}_2 \mathrm{~N}$ and $\mathrm{S}_2 \mathrm{M}$ both make angle $\left(180^{\circ}-\theta\right)$ with the velocity vector $\vec{u}$. The component of observer velocity towards the sources $N$ and $M$ is $u_o=-u \cos \theta$. Apply Doppler's effect equation to get

$ \begin{aligned} \nu_N^{\prime} & =\frac{v+u_o}{v-u_s} \nu_N=\frac{v-u \cos \theta}{v} \nu_N \\\\ & =\left(1-\frac{u}{v} \frac{u t-D}{\sqrt{d^2+(u t-D)^2}}\right) \nu_N, \\\\ \nu_M^{\prime} & =\frac{v+u_o}{v-u_s} \nu_M=\frac{v-u \cos \theta}{v} \nu_M \\\\ & =\left(1-\frac{u}{v} \frac{u t-D}{\sqrt{d^2+(u t-D)^2}}\right) \nu_M . \end{aligned} $

The beat frequency heard by the observer at time $t(\geq$ $\left.t_Q\right)$ is

$ \begin{aligned} \nu(t) & =\nu_N^{\prime}-\nu_M^{\prime} \\\\ & =\left(1-\frac{u}{v} \frac{u t-D}{\sqrt{d^2+(u t-D)^2}}\right)\left(\nu_N-\nu_M\right) ..........(2) \end{aligned} $

Substitute $t=0$ and $t=t_Q={D \over u}$ in equation (1) to get $\nu_P$ and $\nu_Q$ and substitute $t=t_R=2 {D \over u}$ in equation (2) to get $\nu_R$ i.e.,

$ \begin{aligned} & \nu_P=\nu(t=0)=\left(1+\frac{u}{v} \frac{D}{\sqrt{d^2+D^2}}\right)\left(\nu_N-\nu_M\right) \\\\ & \nu_Q=\nu(t=D / u)=\left(\nu_N-\nu_M\right) \\\\ & \nu_R=\nu(t=2 D / u)=\left(1-\frac{u}{v} \frac{D}{\sqrt{d^2+D^2}}\right)\left(\nu_N-\nu_M\right) \end{aligned} $

$ \therefore $ $ \nu_P+\nu_R=2 \nu_Q $

Differentiate equations (1) w.r.t. time t to get rate of change of beat frequency

$ \frac{\mathrm{d} \nu(t)}{\mathrm{d} t}=-\left(\nu_N-\nu_M\right) \frac{u^2}{v} \frac{d^2}{\left(d^2+(D-u t)^2\right)^{3 / 2}} $ ...........(3)

From equation (3), the slope is negative and its magnitude is maximum when $t={D \over u}=t_Q$ (denominator is minimum). Thus the rate of change of beat frequency is maximum when car passes through $Q$.

${f_1} = f\left[ {{v \over {v - {v_s}}}} \right] = f \times {{320} \over {300}}Hz$

${f_2} = f\left[ {{v \over {v + {v_s}}}} \right] = f \times {{320} \over {340}}Hz$

$\left( {{{{f_2}} \over {{f_1}}} - 1} \right) \times 100 = \left( {{{300} \over {340}} - 1} \right) \times 100 = 12\% $

Length of pipe $=85$ $cm$ $=0.85m$

Pipe is closed from one end so it behaves as a closed organ pipe

Frequency of oscillations of air column in closed organ pipe is given by,

$f = {{\left( {2n - 1} \right)\upsilon } \over {4L}}$

$f = {{\left( {2n - 1} \right)\upsilon } \over {4L}} \le 1250$

$ \Rightarrow {{\left( {2n - 1} \right) \times 340} \over {0.85 \times 4}} \le 1250$

$ \Rightarrow 2n - 1 \le 12.5 \approx 6$

Possible value of n = 1, 2, 3, 4, 5, 6

So, number of possible natural frequencies lie below 1250 Hz is 6.

The displacement of a stationary wave is given by

$y(x,t) = A\sin \left( {{{2\pi x} \over \lambda }} \right)\cos (2\pi vt)$.

The boundary conditions give node at x = 0 and anti-node at x = 3 m i.e.,

y(0, t) = 0, ....... (1)

y(3, t) = $\pm$A. ..... (2)


The fundamental frequency is given by

${v_0} = {v \over \lambda } = {v \over {4l}} = {{100} \over {4(3)}} = {{25} \over 3}$ Hz.

Thus, the waveform will satisfy equations (1) and (2), and the permissible frequencies will be odd multiples of v₀.

Minimum length $ = {\lambda \over 4} \Rightarrow \lambda = 4l$

Now, v = f $\lambda$ = (244) $\times$ 4 $\times$ l

as l = 0.350 $\pm$ 0.005

$\Rightarrow$ v lies between 336.7 m/s to 346.5 m/s

Now, $v = \sqrt {{{\gamma RT} \over {M \times {{10}^{ - 3}}}}} $, here M is molecular mass in gram $ = \sqrt {100\gamma RT} \times \sqrt {{{10} \over m}} $.

For monoatomic gas,

$\gamma$ = 1.67

$\Rightarrow$ $v = 640 \times \sqrt {{{10} \over m}} $

For diatomic gas,

$\gamma = 1.4 \Rightarrow v = 590 \times \sqrt {{{10} \over m}} $

$\therefore$ ${v_{Ne}} = 640 \times {7 \over {10}} = 448$ m/s

${v_{Ar}} = 640 \times {{17} \over {32}} = 340$ m/s

${v_{{O_2}}} = 590 \times {9 \over {16}}=331.8$ m/s

${v_{{N_2}}} = 590 \times {3 \over 5} = 354$ m/s

$\therefore$ Only possible answer is Argon.

Fundamental frequency,

$f = {v \over {2\ell }} = {1 \over {2\ell }}\sqrt {{T \over \mu }} = {1 \over {2\ell }}\sqrt {{T \over {A\rho }}} $

$\left[ {\,\,} \right.$ as $v = \sqrt {{T \over \mu }} $ $\,\,\,\,\,\,$ and $\,\,\,\,\,\,$ $\left. {\mu = {m \over \ell }\,\,} \right]$

Also, $Y = {{T\ell } \over {A\Delta \ell }} \Rightarrow {T \over A} = {{Y\Delta \ell } \over \ell }$

$ \Rightarrow f = {1 \over {2\ell }}\sqrt {{{\gamma \Delta \ell } \over {\ell \rho }}} ....\left( i \right)$

Putting the value of $\ell ,{{\Delta \ell } \over \ell },\rho $ $\,\,\,\,\,\,$ and

$\,\,\,\,\,\,$ $\gamma $ in $e{q^n}.\left( i \right)$ we get,

$f = \sqrt {{2 \over 7}} \times {{{{10}^3}} \over 3}$ $\,\,\,\,\,\,$ or, $\,\,\,\,\,\,$ $f \approx 178.2\,Hz$

Doppler's effect gives

${f_2} = \left[ {{{v + {u_0}} \over {v - {u_s}}}} \right]{f_1}$ ..... (1)

Let the wind speed be w and wind moves towards the source in case (i) and towards the observer in case (ii) (see figure).

In the case (i), u₀, the speed of observer w.r.t. medium considered positive when it moves towards the source is +(u $-$ w) and u_s, the speed of source w.r.t. medium considered positive when it moves towards observer is +(u + w). Substitute these values in equation (1) to get

${f_2} = \left[ {{{v + (u - w)} \over {v - (u + w)}}} \right]{f_1}$,

which is greater than f₁ (assuming u > w). In the case (ii), u₀ is +(u + w) and u_s is +(u $-$ w). Substitute these values in equation (1) to get

${f_2} = \left[ {{{v + (u + w)} \over {v - (u - w)}}} \right]f1$,

which is again greater than f₁.

Number of nodes = 6

From the given equation, we can see that

$k = {{2\pi } \over \lambda } = 62.8$ m^$-$1

$\therefore$ $\lambda = {{2\pi } \over {62.8}}$ m = 0.1 m

$1 = {{5\lambda } \over 2}$ = 0.25 m

The mid-point of the string is P, an antinode

$\therefore$ maximum displacement = 0.01 m

$\omega$ = 2$\pi$f = 628 s^$-$1

$\therefore$ $f = {{628} \over {2\pi }} = 100$ Hz

But this is fifth harmonic frequency.

$\therefore$ Fundamental frequency f₀

$ = {f \over 5} = 20$ Hz.

The fundamental frequency of open tube

${v_0} = {v \over {2{l_0}}}\,\,\,\,\,\,\,\,\,\,\,...\left( i \right)$

That of closed pipe

${v_c} = {\upsilon \over {4{l_c}}}\,\,\,\,\,\,\,\,\,\,\,...\left( {ii} \right)$
According to the problem ${l_c} = {{{l_0}} \over 2}$

Thus ${v_c} = {\upsilon \over {{l_0}/2}} \Rightarrow {v_c}{\upsilon \over {2l}}\,\,\,\,...\left( {iii} \right)$

From equations $(i)$ and $(iii)$

${v_0} = {v_c}$

Thus, ${v_c} = f$ $\,\,\,\left( {\,\,} \right.$ as ${v_0} = f$ is given $\left. {\,\,} \right)$

A student is conducting a resonance column experiment. The tube has a diameter of 4 cm, and the tuning fork vibrates at 512 Hz. The air temperature is 38^oC, where the speed of sound is 336 m/s. The zero mark on the meter stick aligns with the top of the tube. For the first resonance, the water level reads :

During the first resonance :

$ \frac{\lambda}{4} = l_1 + e $

where $ l_1 $ is the length of the air column at resonance

Given that :

$ \text{end correction} = e = 0.6r = 0.6 \times 2 = 1.2 \, \text{cm} $

where $ r $ is the radius of the tube

The wavelength $ \lambda $ can be expressed as :

$ \lambda = 4(l_1 + e) $

The frequency $ f $ is related to the wavelength $ \lambda $ by :

$ f = \frac{v}{\lambda} = \frac{v}{4(l_1 + e)} $

Rearranging for $ l_1 $ :

$ 4(l_1 + e) = \frac{v}{f} $

Solving for $ l_1 $ :

$ l_1 = \frac{v}{4f} - e $

Substitute the given values :

$ l_1 = \frac{336 \, \text{m/s}}{4 \times 512 \, \text{Hz}} - 0.012 \, \text{m} $

$ l_1 = \frac{336}{2048} - 0.012 $

$ l_1 = 0.164 \, \text{m} - 0.012 \, \text{m} $

$ l_1 = 0.152 \, \text{m} $

Converting to cm :

$ l_1 = 15.2 \, \text{cm} $

Given wave equation is

$y\left( {x,t} \right){ = _e}\left( { - a{x^2} + b{t^2} + 2\sqrt {ab} \,xt} \right)$

$ = {e^{ - \left[ {{{\left( {\sqrt {ax} } \right)}^2} + {{\left( {\sqrt {bt} } \right)}^2} + 2\sqrt a x.\sqrt b t} \right]}}$

$ = {e^{ - {{\left( {\sqrt a x + \sqrt b t} \right)}^2}}}$

$ = {e^{ - {{\left( {x + \sqrt {{b \over a}} t} \right)}^2}}}$

It is a function of type $y = f\left( {x + vt} \right)$

$ \Rightarrow $ Speed of wave $ = \sqrt {{b \over a}} $

u = 36 km h^$-$1 = 10 ms^$-$1,

$v$ = 320 ms^$-$1, $\nu $ = 8 kHz

The sound reflected from the building may be imagined to be coming from the mirror image. The driver is approaching the image-source which is also approaching him with the same speed. Hence the frequency of sound heard by the driver is

$\nu $' = $\nu $ $\left( {{{v + u} \over {v - u}}} \right)$

= 8 kHz $\times$ $\left( {{{320 + 10} \over {320 - 10}}} \right)$ = 8.5 kHz

(1) For a pipe closed at one end, we have

${{{\lambda _f}} \over 4} = L$ or ${\lambda _f} = 4L$

Therefore, the sound waves are longitudinal.

(2) For a pipe open at both ends, we have

${{{\lambda _f}} \over 2} = L$ or ${\lambda _f} = 2L$

Therefore, the sound waves are longitudinal.

(3) For a stretched wire clamped at both ends, we have

${{{\lambda _f}} \over 2} = L$

Vibration on the string is transverse.

(4) For a stretched wire clamped at both ends and at mid-point, we have

${{{\lambda _f}} \over 2} = {L \over 2} \Rightarrow {\lambda _f} = L$

Therefore, the vibration on the string is transverse.

$y = 0.02\left( m \right)\sin \left[ {2\pi \left( {{t \over {0.04\left( s \right)}}} \right) - {x \over {0.50\left( m \right)}}} \right]$

But $y = a\sin \left( {\omega t - kx} \right)$

$\therefore$ $\omega = {{2\pi } \over {0.04}} \Rightarrow v = {1 \over {0.04}} = 25\,Hz$

$k = {{2\pi } \over {0.50}} \Rightarrow \lambda = 0.5m$

$\therefore$ velocity, $v = v\lambda = 25 \times 0.5\,m/s = 12.5\,m/s$

Velocity on a string is given by

$v = \sqrt {{T \over \mu }} $

$\therefore$ $T = {v^2} \times \mu = {\left( {12.5} \right)^2} \times 0.04 = 6.25\,N$

The fundamental mode in a pipe closed at one end and the second harmonic in a string are shown in the figure.

Fundamental frequency of a hollow pipe closed at one end is

${\upsilon _p} = {v \over {4L}}$

where, v = speed of sound, L = length of a pipe

Frequency of second harmonic of a string is

${\upsilon _s} = {2 \over {2l}}\sqrt {{T \over \mu }} = {1 \over l}\sqrt {{T \over \mu }} $

where, T = tension of the string, m = mass per unit length of the string, l = length of the string

${\upsilon _s} = {1 \over l}\sqrt {{T \over {{m \over l}}}} = {1 \over l}\sqrt {{{Tl} \over m}} = \sqrt {{T \over {ml}}} $ ...... (i)

where m is the mass of the string

According to given problem, ${\upsilon _p} = {\upsilon _s}$

$\therefore$ ${v \over {4L}} = \sqrt {{T \over {ml}}} $ or $m = {{16{L^2}T} \over {{v^2}l}}$

Substituting the given values, we get

$m = {{16 \times {{(0.8)}^2} \times (50)} \over {{{(320)}^2} \times 0.5}} = 0.01$ kg = 10 gram

Maximum number of beats $ = \left( {v + 1} \right) - \left( {v - 1} \right) = 2$

$v_m^2 - {u^2} = 2as \Rightarrow v_m^2 = 2 \times 2 \times s$

$\therefore$ ${v_m} = 2\sqrt s $

According to Doppler's effect

$0.94v = v\left[ {{{330 - 2\sqrt s } \over {330}}} \right] \Rightarrow s = 98.01\,m$

We have,

$\overrightarrow J = \overrightarrow r \times \overrightarrow F $

$|\overrightarrow J | = rF\sin 180^\circ = 0$

Therefore, the angular momentum is conserved, that is,

$mvr\sin \theta = $ Constant

where $\theta$ is the angle between the velocity $\overrightarrow v $ and position vector $\overrightarrow r $ of $-$q with respect to Q.