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(c) is lower than its natural frequency

T_new = √2 T_old.   

A medium in which speed of wave motion is independent of frequency of wave is called non-dispersive medium. For sound, air is non dispersive medium.

To and for motion in the same path is called Oscillation.

Acceleration should be proportional to displacement and always directed towards mean position.

Condition (i) is not sufficient, because direction of acceleration is not mentioned. In S.H.M, the acceleration is always in a direction opposite to that of the displacement.

The necessary and sufficient condition for motion to be S.H. is that the restoring force must be linear, i.e., F = -ky or torque, τ = -cθ.

Acceleration ∝ (-displacement).

The motion which repeats itself over and over again about its mean position such that it remains confined within well defined limits (known as extreme positions) on either sides of the mean position, is called an oscillatory motion.

There will be no effect because the time period does not depend upon the nature of material of the bob.

S.H.M. is the projection of uniform circular motion upon any of the diameters of the circle.

It is the distance traveled by the particle from its mean position at ant instant.

Harmonics are the notes of frequencies which are integral multiple of the fundamental frequency.

Reverberation is the persistence of sound after the source has stopped to produce sound.

The phenomenon of persistence or prolongation of sound after the source has stopped emitting sound is called reverberation. The time for which the sound persists until it becomes inaudible is called the reverberation time.

For an echo of a simple sound to be heard, the minimum distance between the speaker and the walls should be 17 m, so in any room having length less than 17 m, our ears can not distinguish between sound received directly and sound received after reflection.

Let a particle moves with constant angular speed ω along a circular path of radius 'a'. The foot of the perpendicular drawn from the position of the particle on the diameter of circular path executes S.H.M.

When the particle moves from position P₁ to P₂ in time t, the displacement of Q (foot of the perpendicular drawn from the position of the particle on the diameter of the circle) executing SHM from ΔOP₂ is given as

or, OQ = OP₂ sinθ

y = a sinθ

As θ = ωt

The equation of the particle in SHM,

y = a sin ωt

and Velocity, v = dy/dt = d/dt (a sin ωt)

or, v = aw cos ωt

We know, potential energy of a particle executing S.H.M. (i.e., oscillator) at any instant is given by

P.E. = 1/2 Mω²x²   ...(i)

But ω = 2π/T = √{k/M}

or, Mω² = k    ...(ii)

Hence eqn. (i) becomes

P.E. = 1/2 kx²    (iii)

Also, kinetic energy of the particle executing S.H.m is given by

K.E. = 1/2 Mω²(r² - x²)

where r = amplitude of the particle

Using eqn. (ii), we get

K.E. = 1/2 k(r² - x²)    ...(iv)

Thus the total energy at any instant of the particle executing S.H.M. is given by

E = K.E. + P.E. = 1/2 k(r² - x²) + 1/2 kx²

or, E = 1/2 kr²    ....(v)

For a simple harmonic motion, displacement

y = r sin(ωt + ϕ)

Where r is amplitude, ω is angular velocity and ϕ is phase difference

But ω = \(\frac{2\pi}{T}\) = 2πf 

And ϕ = \(\frac{2\pi}{T}x\)

Where λ is wavelength and x is path difference.

∴ y = rsin 2π(\(\frac{t}{T}+\frac{x}{\lambda}\))

When amplitude of the vibrating pendulum is small then θ is small.

Here the restoring force F = -mg sin θ = -mgθ = -mgx/l,

where x is the displacement of bob and l is the length of pendulum.

Hence F ∝ -x. Since F is directed towards mean position.

Therefore, the motion of the bob of simple pendulum will be SHM, if θ is small.

Assume, n₁ > n₁

Frequency of Beat, v_b = n₁ - n₂

∴ Time period, \(T_b=\frac{1}{v_b}\)

= \(\frac{1}{n_1-n_2}s\)

\(v\propto \sqrt T\)

or \(\frac{v_r}{v_0}=\sqrt{\frac{T}{T_0}}\)

From question, v_T = 3v₀ 

∴ \(\frac{3v_0}{v_0}=\sqrt{{\frac{T}{273+0}}}\)

or \(\sqrt T=3\sqrt{ 273}\)

T = 9 × 273 = 2457 K 

T = 2457 – 273 

T = 2184ºC

On reflection from a denser medium, a wave suffers a sudden phase reversal. 

Intensity = amplitude² ∝ 1/(distance)² 

. ^.. Required ratio = y/x