HC Verma solutions for Concepts of Physics Vol. 1 [English] Class 11 and 12 chapter 1 - Introduction to Physics [Latest edition]

The metre is defined as the distance travelled by light in `1/(299,792,458)` second. Why didn't people choose some easier number such as  `1/(300,000,000)` second? Why not 1 second?

What are the dimensions of volume of a cube of edge a.

What are the dimensions of volume of a sphere of radius a?

What are the dimensions of the ratio of the volume of a cube of edge a to the volume of a sphere of radius a?

Suppose you are told that the linear size of everything in the universe has been doubled overnight. Can you test this statement by measuring sizes with a metre stick? Can you test it by using the fact that the speed of light is a universal constant and has not changed? What will happen if all the clocks in the universe also start running at half the speed?

If all the terms in an equation have same units, is it necessary that they have same dimensions? If all the terms in an equation have same dimensions, is it necessary that they have same units?

If two quantities have same dimensions, do they represent same physical content?

It is desirable that the standards of units be easily available, invariable, indestructible and easily reproducible. If we use foot of a person as a standard unit of length, which of the above features are present and which are not?

Suggest a way to measure the thickness of a sheet of paper.

Suggest a way to measure the distance between the sun and the moon.

Which of the following sets cannot enter into the list of fundamental quantities in any system of units?

A physical quantity is measured and the result is expressed as nu where u is the unit used and n is the numerical value. If the result is expressed in various units then 

Suppose a quantity x can be dimensionally represented in terms of M, L and T, that is, `[ x ] = M^a L^b T^c`.  The quantity mass

A dimensionless quantity

A unitless quantity

\[\int\frac{dx}{\sqrt{2ax - x^2}} = a^n \sin^{- 1} \left[ \frac{x}{a} - 1 \right]\] 
The value of n is

The dimensions ML⁻¹ T⁻² may correspond to

Choose the correct statements(s):

Choose the correct statements(s):
(a) All quantities may be represented dimensionally in terms of the base quantities.
(b) A base quantity cannot be represented dimensionally in terms of the rest of the base quantities.
(c) The dimensions of a base quantity in other base quantities is always zero.
(d) The dimension of a derived quantity is never zero in any base quantity.

Find the dimensions of linear momentum . 

Find the dimensions of frequency .

Find the dimensions of pressure.

Find the dimensions of
(a) angular speed ω,
(b) angular acceleration α,
(c) torque τ and
(d) moment of interia I.
Some of the equations involving these quantities are \[\omega = \frac{\theta_2 - \theta_1}{t_2 - t_1}, \alpha = \frac{\omega_2 - \omega_1}{t_2 - t_1}, \tau = F . r \text{ and }I = m r^2\].
The symbols have standard meanings.

Find the dimensions of electric field E. 

The relevant equations are \[F = qE, F = qvB, \text{ and }B = \frac{\mu_0 I}{2 \pi a};\]
where F is force, q is charge, v is speed, I is current, and a is distance.

Find the dimensions of magnetic field B.
The relevant equation are \[F = qE, F = qvB, \text{ and }B = \frac{\mu_0 I}{2 \pi a};\]

Find the dimensions of magnetic permeability \[\mu_0\] 
The relevant equation are \[F = qE, F = qvB, \text{ and }B = \frac{\mu_0 I}{2 \pi a};\]

Find the dimensions of electric dipole moment p .
The defining equations are p = q.d and M = IA;
where d is distance, A is area, q is charge and I is current.

Find the dimensions of magnetic dipole moment M.
The defining equations are p = q.d and M = IA;
where d is distance, A is area, q is charge and I is current.

Find the dimensions of Planck's constant h from the equation E = hv where E is the energy and v is the frequency.

Find the dimensions of the specific heat capacity c.
(a) the specific heat capacity c,
(b) the coefficient of linear expansion α and
(c) the gas constant R.
Some of the equations involving these quantities are \[Q = mc\left( T_2 - T_1 \right), l_t = l_0 \left[ 1 + \alpha\left( T_2 - T_1 \right) \right]\] and PV = nRT.

Taking force, length and time to be the fundamental quantities, find the dimensions of density .

Taking force, length and time to be the fundamental quantities, find the dimensions of pressure .

Taking force, length and time to be the fundamental quantities, find the dimensions of momentum. 

Taking force, length and time to be the fundamental quantities, find the dimensions of energy.

Suppose the acceleration due to gravity at a place is 10 m/s². Find its value if cm/(minute)².

The average speed of a snail is 0 . 020 miles/ hour and that of a leopard is 70 miles/ hour. Convert these speeds in SI units.

The height of mercury column in a barometer in a Calcutta laboratory was recorded to be 75 cm. Calculate this pressure in SI and CGS units using the following data : Specific gravity of mercury = \[13 \cdot 6\] , Density of \[\text{ water} = {10}^3 kg/ m^3 , g = 9 \cdot 8 m/ s^2\] at Calcutta. Pressure
= hpg in usual symbols.

Express the power of a 100 watt bulb in CGS unit.

The normal duration of I.Sc. Physics practical period in Indian colleges is 100 minutes. Express this period in microcenturies. 1 microcentury = 10⁻⁶ × 100 years. How many microcenturies did you sleep yesterday?

The surface tension of water is 72 dyne/cm. Convert it in SI unit.

The kinetic energy K of a rotating body depends on its moment of inertia I and its angular speedω. Assuming the relation to be \[k = KI^0w^B\]  where k is a dimensionless constant, find a and b. Moment of inertia of a sphere about its diameter is  \[\frac{2}{5}M r^2\] 

Theory of relativity reveals that mass can be converted into energy. The energy E so obtained is proportional to certain powers of mass m and the speed c of light. Guess a relation among the quantities using the method of dimensions.

Let I = current through a conductor, R = its resistance and V = potential difference across its ends. According to Ohm's law, product of two of these quantities equals the third. Obtain Ohm's law from dimensional analysis. Dimensional formulae for R and V are \[{\text{ML}}^2 \text{I}^{- 2} \text{T}^{- 3}\] and \[{\text{ML}}^2 \text{T}^{- 3} \text{I}^{- 1}\] respectively.

The frequency of vibration of a string depends on the length L between the nodes, the tension F in the string and its mass per unit length m. Guess the expression for its frequency from dimensional analysis.

Test if the following equation is dimensionally correct:
\[h = \frac{2S cos\theta}{\text{ prg }},\]
where h = height, S = surface tension, ρ = density, I = moment of interia.

Test if the following equation is dimensionally correct:
\[v = \sqrt{\frac{P}{\rho}},\]

where v = velocity, ρ = density, P = pressure

Test if the following equation is dimensionally correct:
\[V = \frac{\pi P r^4 t}{8 \eta l}\]

where v = frequency, P = pressure, η = coefficient of viscosity.

Test if the following equation is dimensionally correct:
\[v = \frac{1}{2 \pi}\sqrt{\frac{mgl}{I}};\] 
where h = height, S = surface tension, \[\rho\] = density, P = pressure, V = volume, \[\eta =\] coefficient of viscosity, v = frequency and I = moment of interia.

Let x and a stand for distance. Is
\[\int\frac{dx}{\sqrt{a^2 - x^2}} = \frac{1}{a} \sin^{- 1} \frac{a}{x}\] dimensionally correct?