JEE Advanced (One or More Correct Option): Simple Harmonic Motion (Oscillation) | Chapter-wise Tests for JEE Main & Advanced PDF Download

Q.1. One end of a taut string of length 3 m along the x-axis is fixed at x= 0. The speed of the waves in the string is 100 ms^-1. The other end of the string is vibrating in they-direction so that stationary waves are set up in the string. The possible waveform (s) of these stationary waves is(are)
(a)
(b)
(c)
(d)

Correct Answer is option (a, c and d)
Clearly in the given situation a displacement node is present at x = 0 and a displacement antinode is present at x = 3 m.
Therefore, y = 0 at x = 0 and y =  ± A at x = 3 m.
The velocity v = ω/k —6) = 100 ms^-1.
a, c and d are the correct options which satisfy the above conditions.

Q.2. Two vehicles, each moving with speed u on the same horizontal straight road, are approaching each other. Wind blows along the road with velocity w. One of these vehicles blows a whistle of frequency f₁. An observer in the other vehicle hears the frequency of the whistle to be f₂. The speed of sound in still air is V. The correct statement(s) is (are)
(a) If the wind blows from the observer to the source, f₂ > f₁
(b) II the wind blows from the source to the observer, f₂ > f₁
(c) I2 wind blows from observer to the source, f₂ < f₁
(d) It the wind blows from the source to the observer, f₂ < f₁

Correct Answer is option (a and b)

Hence, a and b Correct options.

Q.3. A person blows into open-end of a long pipe. As a result, a high pressure pulse of air travels down the pipe. When this pulse reaches the other end of the pipe.
(a) a high-pressure pulse starts travelling up the pipe, if the other end of the pipe is open.
(b) a low-pressure pulse starts travelling up the pipe, if the  other end of the pipe is open.
(c) a low-pressure pulse starts travelling up the pipe, if the  other end of the pipe is closed.
(d) a high-pressure pulse starts travelling up the pipe, if the other end of the pipe is closed.

Correct Answer is option (b and d)
At open end phase of pressure wave charge by π so compression returns as rare fraction. While at closed end phase of pressure wave does not charge so compression return as compression.

Q.4. Standing waves can be produced.
(a) on a string clamped at both the ends.
(b) on a string clamped at one end free at the other
(c) when incident wave gets reflected from a wall
(d) when two identical waves with a phase difference of it are moving in the same direction

Correct Answer is option (a, b and c)
Standing waves are produced by two similar waves superposing while travelling in opposite direction. This can happen in case (a), (b) and (c).

Q.5.  In a wave motion y = α sin (kx - ωt), y can represent
(a) electric field
(b) magnetic field
(c) displacement
(d) pressure

Correct Answer is option (a, b, c and d)
In the wave motion y = α(kx - ωt), y can represent electric and magnetic fields in electromagnetic waves and displacement and pressure in sound waves.

Q.6. Two identical straight wires are stretched so as to produce 6 beats per second when vibrating simultaneously. On changing the tension slightly in one of them, the beat frequency remains unchanged. Denoting by T₁ , T₂ the higher and the lower initial tension in the strings, then it could be said that while making the above changes in tension.
(a) T₂ was decreased
(b) T₂ was increased
(c) T₁ was decreased
(d) T₁ was increased 

Correct Answer is option (a, b, c and d)
∴ f ∝ √T
Given that T₁ > T₂ ∴ f₁ ∝ f₂
Initially beat frequency (f₁ - f₂) = 6.

Q.7. Velocity of sound in air is 320 m/s. A pipe closed at one end has a length of I m. Neglecting end corrections, the air column in the pipe can resonate for sound of frequency :
(a) 80 Hz
(b) 240 Hz
(c) 320 Hz
(d) 400 Hz

Correct Answer is option (a, b and d)

∴ ....= 80, 240, 400....

Q.8. An organ pipe P₁ closed at one end vibrating in its first harmonic and another pipe P₂ open at ends vibrating in its third harmonic are in resonance with a given tuning fork. The ratio of the length of P₁ to that of P₂ is
(a) 8/3
(b) 3/8
(c) 1/6
(d) 1/3

Correct Answer is option (a, b and d) 

Q.9.  An air column in a pipe, which is closed at one end, will be in resonance with a vibrating tuning fork of frequency 264 Hz if the length of the column in cm is :
(a) 31.25
(b) 62.50
(c) 93.75
(d) 125 

Correct Answer is option (a and c)
The wavelengths possible in an air column in a pipe which has one closed end is

v = 264 Hz as it is in resonance with a vibrating turning fork of frequency 264 Hz.

For n = 1, l = 0.3125 m = 31.25 cm
For  n = 2, l = 0.9375 m = 93.75 cm

Q.10. The displacement of particles in a string stretched in the x-direction is represented by y. Among the following expressions for y, those describing wave motion are:
(a) cos kx sin ωt
(b) k²x² - ω²t²
(c) cos²(kx + ωt)
(d) cos (k²x²- ω²t²)

Correct Answer is option (a and c)
For wave motion, the differential equation is
 ...(i)
NOTE : The wave motion is conditions characterized by the two conditions
f (x, t) = f (x, t + T) ...(ii)
f (x, t) = f (x + λ, t)  ...(iii)