Correct Answer – Option 2 : 2.0 × 10^-24 kg – m/s

Concept:

The momentum, p of an object is defined as the product of its mass, m and velocity, v.

p = mv

Momentum has both direction and magnitude. Its direction is the same as that of velocity, v. The SI unit of momentum is kilogram-metre per second.

Calculation:

Linear momentum = mv = 9.1 × 10^-31 × 2.2 × 10⁶ = 2.0 × 10^-24 kg-m/s

Correct Answer – Option 1 : 344 m/sec

The correct answer is 344 m/sec.

• The speed of sound is the distance traveled per unit of time by a sound wave as it propagates through an elastic medium.
• At 20 °C (68 °F), the speed of sound in air is about 344 meters per second.
• It depends strongly on temperature as well as the medium through which a sound wave is propagating.
• The speed of sound in an ideal gas depends only on its composition and temperature. The speed has a weak dependence on pressure and frequency in ordinary air, deviating slightly from ideal behavior.

Correct Answer – Option 2 : 4260 ± 80 cm³

Concept:

Volume of a cylinder of radius ‘r’ and height h is given by

V = πr² h

Or

\(V = \pi {\left( {\frac{D}{2}} \right)^2}h\)

\(\left[\because {d = 2r \Rightarrow r = \frac{d}{2}} \right]\) 

\(V = \frac{1}{4}\left( {\pi {D^2}h} \right)\)

Where D is the diameter of circular surface

Calculation:

Given,

Diameter of the cylinder, D = 12.6 ± 0.1 cm

Height of the cylinder, h = 34.2 ± 0.1 cm

Here, D = 12.6 cm and h = 34.2 cm

\(V = \frac{\pi }{4} \times {\left( {12.6} \right)^2} \times \left( {34.2} \right)\)

\(V = \frac{\pi }{4} \times 158.76 \times 34.2\)

\(V = \frac{\pi }{4} \times 5429.592\)

V = 1357.398 × π

V = 1357.398 × 3.14

V = 4262.22 cm³

V = 4260 (in three significant numbers)

We know that,

ΔD = ± 0.1 and Δh = ± 0.1

Now, error calculation can be done as

\(\frac{{{\rm{\;\Delta }}V}}{V} = 2\left( {\frac{{{\rm{\Delta }}D}}{D}} \right) + \frac{{{\rm{\Delta }}h}}{h} = \frac{{2 \times 0.1}}{{12.6}} + \frac{{0.1}}{{34.2}}\) 

\(\frac{{{\rm{\Delta }}V}}{V} = 0.0158 + 0.0029\) 

\(\frac{{{\rm{\Delta }}V}}{V} = 0.01879\) 

ΔV = (0.01879) × (4262.22)

ΔV = 79.7 ≈ 80 cm³

∴ For proper significant numbers, volume reading will be V = 4260 ± 80 cm³

Therefore, the value of its volume in appropriate significant figures is 4260 ± 80 cm³

Correct Answer – Option 2 : N1 = 3, N2 = 1, N₃ = 1

Explanation:

As the levels are non-degenerate, there is only one state for each energy.

Let the number of particles occupying the 3 energy states be N1, N2, and N3 respectively

Where 

N₁ + N₂ + N₃ = 5
 

The particles are identifiable, the number of ways of choosing the particles is
\(W = \frac{{5!}}{{{N_1}!\;{N_2}!{N_3}!}}\)

The energy of the system is

0 × N₁ + E × N₂ + 2 E × N₃ = 3E (given)

N2 + 2 N3 = 3  —–(1)
 

Now, the most probable distribution is the one in which W is a maximum, subject to constraint given by equation (1)

Thus if

N₂ = 1

N₃ = (3 -1) /2 = 1

and

N₁ = 5 – (N₁ + N₂) = 5 – 2 = 3

If N2 = 3, N3 = 0

and N1 = 5 – (3 + 0) = 2

No other distribution are possible

For

N₁ = 3, N₃ = 1 and N₃ = 1  

\(W = \frac{{5!}}{{3!}} = 20\)

So must probable distribution is 

N₁ = 3, N₂ = 1 and N₃ = 1

The correct answers are: 

N1 = 3, N2 = 1 and N3 = 1