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Questions:

Generic 2

14 x 2 = 28

You have learnt that a travelling wave in one dimension is represented by a function y = f(x,t) where, x and t must appear in the combination x- vt or x + vt, i.e. y = f( x \(\pm \) vt ). Is the converse true?
(a) Examine if the following functions for Y can possibly represent a travelling wave
(b) Examine if the following functions for Y can possibly represent a travelling wave (x - vt)²

You have learnt that a travelling wave in one dimension is represented by a function y = f (x, t) where x and t must appear in the combination x – v t or x + v t, i.e. y = f (x ± v t). Is the converse true? Examine if the following functions for y can possibly represent a travelling wave :
(i) (x - vt)²
(ii) log [(x + vt)/x₀]
(iii) 1/(x + vt)

A sitar wire is replaced by another wire of same length and material but of three times the earlier radius. If the tension in the wire remains the same, then by what factor will the frequency change?

At what temperature (in ⁰C) Will be speed of sound air be 3 times its value at 0⁰ C?

Equation of a plane progressive wave is given by y = 0.6 sin\(2\pi \left( t-\frac { x }{ 2 } \right) \) On reflection from a denser medium, its amplitude becomes 2/3 of the amplitude of incident wave. What will be equation of reflected wave?

A tuning fork A, marked 512 Hz, produces 5 beats per sec, where sounded with another unmarked tuning fork B. If B is loaded with wax, the number of beats is again 5 per sec. What is the frequency of the tuning fork B when not loaded?

We have 0.5 g of hydrogen gas in a cubic chamber of size 3 cm kept at NTP. The gas in the chamber is compressed keeping the temperature constant till a final pressure of 100 atm. Is one justified in assuming the ideal gas law, in the final state? (Hydrogen molecules can be consider as spheres of radius 1 \(\overset { o }{ A }\)).

In a refrigerator, one removes heat from a lower temperature and deposits to the surroundings at a higher temperature. In this process, mechanical work has to be done, which is provided by an electric motor. If the motor is of 1kW power and heat is transferred from -3°C to 27°C. Find the heat taken out of the refrigerator per second assuming its efficiency is 50% of a perfect engine.

A person of mass 60 kg wants to lose 5 kg by going up and down a 10 m high stairs. Assume he burns twice as much fat while going up than coming down. If 1kg of fat is burnt on expending 7000 kcal calories, how many times must he go up and down to reduce his weight by 5 kg?

10.

Equation of a plane progressive wave is given by y = 0.6 sin\(2\pi \left( t-\frac { x }{ 2 } \right) \)
On reflection from a denser medium, its amplitude becomes 2/3 of the amplitude of incident wave. What will be equation of reflected wave?

11.

A solid cylinder of mass 20 kg rotates about its axis with angular speed of 100 rad/s. The radius of cylinder is 0.25m. What is KE of rotation of cylinder?

12.

Why does a solid sphere have smaller moment of inertia than a hollow cylinder of same mass and radius about an axis passing through their axis of symmetry?

13.

Calculate the work done by a car against gravity is zero because force of gravity is vertical and motion of car is along a straight horizontal road.

14.

A person of mass 50 kg stands on a weighing scale on a lift. If the lift is descending with a downwards acceleration of 9 m/s² , what would be the reading of the weighing scale ?
When a lift descends with a downward acceleration 8, V the apparent weight of a body of mass m is given by w' = R = m (g - a)

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Answers Key:

Generic 2

14 x 2 = 28

(a) Conceptual question based on fundamentals of characteristics of travelling wave.
The converse is not true means if the function can be represented in the form y = f( x \(\pm \) vt ), it does not necessarily express a travelling wave. As the essential condition for a travelling wave is that the vibrating particle must have finite displacement value for all x and t.
\(\frac { 1 }{ x+vt } \)
\(\\ For\ x=0,\ t=0,\ f(x)\rightarrow \infty \)
Though the function is of (x±vt) type still at x=0, it is infinite, hence, it is not a wave.
(b) Conceptual question based on fundamentals of characteristics of travelling wave.
The converse is not true means if the function can be represented in the form y = f( x ± vt ), it does not necessarily express a travelling wave. As the essential condition for a travelling wave is that the vibrating particle must have finite displacement value for all x and t.
For x = 0,
If t →0, then (x-vt)²→0 which is finite, hence, it is a wave as it passes the two tests.

Conceptual question based on fundamentals of characteristics of travelling wave.
The converse is not true means if the function can be represented in the form y = f( x \(\pm \) vt ), it does not necessarily express a travelling wave. As the essential condition for a travelling wave is that the vibrating particle must have finite displacement value for all x and t.
(i) For x = 0
If t \(\rightarrow \)0, then (x - vt)²\(\rightarrow \)0 which is finite, hence, it is a wave as it passes the two tests.
(ii) log \(\left( \frac { x+vt }{ x_{ 0 } } \right)\)
l\( \\ At\ x=0\ and\ t=0,\)
\( f(x,t)=log\left( \frac { 0+0 }{ x_{ 0 } } \right) \)
= log 0 \(\rightarrow\) not defined
Hence, it is not a wave.
(iii) \(\frac { 1 }{ x+vt } \)
\( \\ For\ x=0,\ t=0,\ f(x)\rightarrow \infty\)
Though the function is of (x\(\pm\) vt) type still at x = 0, it is infinite, hence, it is not a wave.

\({ v }_{ 1 }=\frac { 1 }{ { l }_{ 1 }{ D }_{ 1 } } \sqrt { \frac { { T }_{ 1 } }{ { \pi \rho }_{ 1 } } } \)
Where, D = diameter of wire
\( { v }_{ 2 }=\frac { 1 }{ { l }_{ 2 }{ D }_{ 2 } } \sqrt { \frac { { T }_{ 2 } }{ { \pi \rho }_{ 2 } } } \)
\( { l }_{ 1 }={ l }_{ 2 },\ { \rho }_{ 2 }={ \rho }_{ 1 }\)
\( { T }_{ 2 }={ T }_{ 1 },\ { D }_{ 2 }={ 3D }_{ 1 }\)
\(\\ \Rightarrow { V }_{ 2 }=\frac { { V }_{ 1 } }{ 3 } \)
New frequency is \(\frac { 1 }{ 3 } \) rd of the original frequency.

We know that, speed, \(v\infty \sqrt { T } \)
By formula v = \(\frac { xRT }{ p } \)
Where T is in kelvin
\(\frac { v_{ t } }{ v_{ 0 } } =\sqrt { \frac { 273+t }{ 273+0 } } =3\)
\(\\ \Rightarrow \frac { 273+t }{ 273 } =9\Rightarrow \ t=9\times 273-273=2184^{ 0 }C\)

On reflection from the denser medium, there will be a phase change of 180⁰
Net amplitude = \(\frac { 2 }{ 3 } \times 0.6=0.4\)
Hence, equation of reflected wave will be
y = 0.4sin\(2\pi \left[ t+\frac { x }{ 2 } +\pi \right] \)
\( =0.4sin2\pi (t+{ x }/{ 2 })\)

Frequency of A, v₀= 512 Hz
Number of beats/s = 5
Frequency of B = 512\(\pm \) 5 = 517 or 517 Hz
On loading its frequency decreases from 571 to 507, so that number of beats/s remain 5.
Hence, frequency of B when not loaded = 517Hz.

We have , 0.25 x 6 x 10²³ molecules, each of volume
Molecular volume = 2.5 x 10^-7 m³
Supposing, ideal gas law is valid.
Final volume = \(\frac { { V }_{ in } }{ 100 }\) = \(\frac { { (3) }^{ 3 }\times { 10 }^{ -6 } }{ 100 }\)
\(\approx\) 2.7\(\times\) 10 ^-7 m³
Which is about the molecular volume. hence, intermolecular forces cannot be neglected. Therefore, the ideal gas situation does not hold.

Given, T₁= -3⁰C = - 3 + 273 = 270K
T₂= 27⁰C = 27 + 273 = 300K
Efficiency,
\(\eta =1-\frac { { T }_{ 1 } }{ { T }_{ 2 } } =1-\frac { 270 }{ 300 } =\frac { 1 }{ 10 } \)%
\(or\quad \frac { W }{ Q } =0.5\quad \eta =\frac { 1 }{ 20 }\)
or Q = 20 W = 20kJ per second

Here, m = 60kg, g = 10m/s², h = 10m
In going up and down once, number of kilocalories burnt
= (mgh + mgh/2) =\(\frac { 3 }{ 2 } \) mgh
\(=\frac { 3 }{ 2 } \times \frac { 60\times 10\times 10 }{ 4.2\times 1000 } =\frac { 15 }{ 7 } kcal\)
Total number of kilocalories to be burnt for losing 5 kg of weight = 5\(\times \)7000 = 35000 kcal
\(\therefore \) Number of times of the person has to go up and down the stairs
= \(\frac { 35000 }{ 15/7 } =\frac { 35\times 7 }{ 15 } \times { 10 }^{ 3 }\) = 16.3 x 10³ times

10.

11.

M = 20 Kg, \(\omega =100rad/s,R=0.25m\).
Moment of inertia of cylinder about its own axis
\(=\frac { 1 }{ 2 } MR^{ 2 }=\frac { 1 }{ 2 } \times 20\times { \left( 0.25 \right) }^{ 2 }\)
Rotational KE \(=\frac { 1 }{ 2 } { I\omega }^{ 2 }\)
\(=\frac { 1 }{ 2 } { I\omega }^{ 2 }=\frac { 1 }{ 2 } \times 0.625\times { \left( 100 \right) }^{ 2 }=3125J\).

12.

All mass of a hollow cylinder at a distance R from axis of rotation. Whereas in case of a sphere, most of mass lies at a distance less than R from axis of rotation. As moment of inertia is \(\sum M_{ i }{ R }_{ i }^{ 2 }\), so sphere as a lower value of moment of inertia.

13.

The workdone by a car against gravity is zero because force of gravity is vertical and motion of car is along a straight horizontal road. As angle \(\theta\) between directions of force and displacement is 90⁰, hence work done is zero.

14.

When a lift descending with a downward acceleration a, the apparent weight of a body of mass m is given by w' = R = m (g - a)
Given, Mass of the person, m = 50 kg
Descending acceleration, a = 9 m/s²
Acceleration due to gravity, g = 10 m/s²
Apparent weight of the person,
R = m ( g - a ) = 50 ( 10 - 9 ) = 50 N
\(\therefore \) Reading of the weighing scale = \(\frac { R }{ g } \) = \(\frac { 50 }{ 10 } \) = 5 kg