Explore topic-wise InterviewSolutions in .

(i) When the spring is pulled sideways, the kink moves at 90º to the length of the spring. Waves are transverse.

(ii) Waves in this case are longitudinal, because molecules of the liquid will move along the direction of motion of the piston.

(iii) The water surface is cut laterally and pushed backwards by the propeller of motion boat. Therefore, the waves are a mixture of longitudinal and transverse waves.

(iv) Light waves (from sun to earth) are electromagnetic waves which are transverse in nature.

(v) Ultrasonic waves in air are basically sound waves of frequency greater than the audible frequencies. They are, therefore, longitudinal waves.

As 2sin A cos B = sin(A + B) + sin(A – B)

∴  y = 2Asin kx cos ωt

= Asin(kx + ωt) + Asin(kx − ωt)

According to superposition principle,

y = y₁ + y_2, 

And  y₁= A sin (ωt – kx)

= −A sin (kx – ωt)

y₂ = y − y₁ = 2Asin kx cos ωt  + Asin(kx − ωt)

= Asin(kx − ωt) + 2Asin(kx − ωt)

= Asin(kx − ωt) − 2Asin(kx − ωt)

Characteristics of standing waves : 

(1) The disturbance confined to a particular region 

(2) There is no forward motion of the disturbance beyond this particular region. 

(3) The total energy is twice the energy of each wave. 

(4) Points of zero amplitude are known as nodes.

The distance between two consecutive nodes is λ/2.

(5) Points of maximum amplitude is known as antinodes. The distance between two consecutive antinodes is also λ/2. The distance between a node and adjoining antinode is λ/4. 

(6) The medium splits up into a number of segments. 

(7) All the particles in one segment vibrate in the same phase. Particles in two consecutive segments differ in phase by 180°. 

(8) Twice during each vibration, all the particles of the medium pass simultaneously through their mean position.

The important characteristics of wave motion:

(i) Wave motion is a sort of disturbance which travels through a medium.

(ii) A material medium is essential for the propagation of mechanical waves. The medium must possess three properties, viz., elasticity, inertia and minimum friction amongst the particles of the medium.

(iii) The velocity of the particles during their vibration is different at different position.

(iv) Energy is propagated along with the wave motion without any net transported of the medium.

(v) There is a continuous phase difference amongst successive particles of the medium.

Liquids and gases cannot sustain shearing stress. Therefore transverse waves in the form of crests and troughs (involving change of shape) are not possible in fluids. Rather, the fluid posses volume elasticity. Therefore, compressions and rarefactions (involving changes in volume) can be propagated through fluids

When a particle performs linear SHM, the force acting on the particle is always directed towards the mean position. The magnitude of the force is directly proportional to the magnitude of the displacement of the particle from the mean position. Thus, if \(\vec F\) is the force acting on the particle when its displacement from the mean position is \(\vec x\), \(\vec F\) = - k\(\vec x\).....(1)

where the constant k, the force per unit displacement, is called the force constant. The minus sign indicates that the force and the displacement are oppositely directed. The velocity of the particle is \(\frac{d\vec x}{dt}\) and its acceleration is \(\frac{d^2\vec x}{dt^2}\).

Let m be the mass of the particle. 

Force = mass × acceleration

∴ \(\vec F\) m \(\frac{d^2\vec x}{dt^2}\)

Hence, from Eq. (1),

m\(\frac{d^2\vec x}{dt^2}\) = -k\(\vec x\)

∴ \(\frac{d^2\vec x}{dt^2}\) + \(\frac km\vec x\) = = 0 … (2)

This is the differential equation of linear SHM.

Potential energy of particle at any instant t second is E_p = \(\frac{1}{2}\) m ω² y².

Kinetic energy of particle at any instant t second is, EK = \(\frac{1}{2}\) m ω² (A² – y²).

Total energy of particle at any instant is constant E = E_K + E_p = \(\frac{1}{2}\)m ω² A² .

A is the amplitude and w is the angular frequency and m is mass of the particle y is the displacement of a particle from its mean position in t seconds.

x = a cos ωt + b sin ωt

Now, \(\frac{dx}{dt}\) = – aω sinωt + bω cosωt

\(\frac{d^2x}{dt^2}\) = – aω² cos ωt + bω² sin ωt

⇒ \(\frac{d^2x}{dt^2}\) = – ω² (a cos ωt+ b sin ωt)

⇒ \(\frac{d^2x}{dt^2}\) = – ω²x

⇒ α = – ω²x

Hence an SHM

For a SHM PE = \(\frac{1}{2}\) kx²

= \(\frac{1}{2}\) mω² x² = \(\frac{1}{2}\)mω A² sin² ωt

KE= \(\frac{1}{2}\) mv²

v =\(\frac{dx}{dt}\) = – Aω cos ωt

⇒ KE = \(\frac{1}{2}\) mA²ω² cos² ωt

Total energy = KE + PE = \(\frac{1}{2}\)mω² A² cos² ωt + \(\frac{1}{2}\)mω² sin² ωt 

Total energy = \(\frac{1}{2}\) mω² A².

Data : T = 2 s, g = 9.81 m/s²

Period of a simple pendulum, T = 2π \(\sqrt{\frac{L}{g}}\)

For a seconds pendulum, 2 = 2π \(\sqrt{\frac{L}{g}}\)

∴ The length of the seconds pendulum,

L = \(\frac{g}{\pi^2}\) = \(\frac{9.81}{(3.142)^2}\) = 0.9937

(d) 2

The time period of the shorter pendulum is half that of the longer pendulum. Therefore, the pendulums will again be in phase ( at the mean position). When the shorter pendulum has completed 2 oscillations.

T = 2\(\pi\) \(\sqrt{\frac Lg}\)

\(\therefore\) \(\frac{T_2}{T_1}\) = \(\sqrt{\frac {L_2}{2}}\) 

\(\therefore\) \(\frac{T_2}{2}\)= √2 

\(\therefore\) T₂ : 2√2 s giuves the required period

(1) Seconds pendulum: A simple pendulum of period two seconds is called a seconds pendulum.

(2) The period of a simple pendulum is

T = 2π \(\sqrt{\frac Lg}\)

For a seconds pendulum, T = 2s.

∴ 2 = 2π\(\sqrt{\frac Lg}\)

∴ L = \(\frac{g}{π^2}\)

This expression gives the length of the seconds pendulum at a place where acceleration due to gravity is g.

(3) At a given place, the value of g is constant. 

∴ L = g/π² = a fixed value, at a given place.

[Note : Because the effective gravitational acceleration varies from place to place, the length of a seconds pendulum should be changed in direct proportion. Since the effective gravitational acceleration increases from the equator to the poles, so should the length of a seconds pendulum be increased.]

T = 2\(\pi\) \(\sqrt{\frac Lg}\)

\(\therefore\) \(\frac{T_2}{T_1}\) = \(\sqrt{\frac {L_2}{L_1}}\) = √9 = 3

\(\therefore\) T₁ : T₂ = 1 : 3

Acceleration due to gravity inside Earth

g′ = \(\frac{GM}{R^3}x=\frac{g}{R}x\) 

Here x = distance of the point from centre of earth (x<R)

If, block of mass m is placed along the diameter inside the earth.

So, force on block

F = − \(\frac{mg}{R}x\) = −kx

∴ T = 2π\(\sqrt{\frac{m}{k}}\) = 2π\(\sqrt{\frac{m}{\frac{mg}{R}}}\) 

T = 2π\(\sqrt{\frac{R}{g}}\)

Velocity of sound in air increases by 0.61 m/s, when temperature rises by 1ºC.

Wavelength of visible light is of the order of 6 × 10^-7m. wavelength of audible sound in air varies from 16.6 × 10^-3 m to 16.6 m.

This because velocity of sound in air is only 331 m/s, whereas velocity of light is 3 × 10⁸ m/s. Hence the flash is seen much before the thunder of lightening is heard.

This is because not material medium is present over a long distance between earth and other planets.