Table of Contents
QUESTIONS FOR SHORT ANSWER
1. 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 ?
To maintain the historical length of the metre and ensure continuity.
Explanation:
The value 299,792,458 was selected so that the new definition of the metre would exactly match the previously established standard. A simpler number or 1 second would change the actual length of a metre, making past measurements and instruments inconsistent.
2. What are the dimensions of : (a) volume of a cube of edge a, (b) volume of a sphere of radius a, (c) the ratio of the volume of a cube of edge a to the volume of a sphere of radius a ?
(a) \(L^3\), (b) \(L^3\), (c) Dimensionless \((M^0 L^0 T^0)\).
Explanation:
Volume is the product of three lengths. For cube \((a^3)\) and sphere \(\left(\frac{4}{3}\pi a^3\right)\), dimensions are \(L^3\). Ratio of volumes \(\left(\frac{L^3}{L^3}\right)\) is dimensionless.
3. 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 ?
No for the metre stick; Yes for the speed of light; No if clocks also slow down.
Explanation:
A metre stick also doubles, so no change is detected. Speed of light allows detection since time doubles for distance. If clocks run at half speed, time appears unchanged, making change undetectable.
4. 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 ?
Yes for dimensions; No for units.
Explanation:
Same units imply same dimensions. Same dimensions do not imply same units, as quantities can be expressed in different unit systems.
5. If two quantities have same dimensions, do they represent same physical content ?
No.
Explanation:
Same dimensions do not imply same physical meaning. Example: work and torque have dimensions \(ML^2T^{-2}\) but represent different concepts.
6. 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 ?
Availability and reproducibility are present; invariability and indestructibility are absent.
Explanation:
Foot is available and reproducible but not invariable (size varies) and not indestructible.
7. Suggest a way to measure : (a) the thickness of a sheet of paper, (b) the distance between the sun and the moon.
(a) Measuring a stack; (b) The parallax method.
Explanation:
Thickness is measured by stacking many sheets and dividing. Distance is measured using parallax and trigonometry.
OBJECTIVE I
1. Which of the following sets cannot enter into the list of fundamental quantities in any system of units ?
(a) length, mass and velocity
(b) length, time and velocity
(c) mass, time and velocity
(d) length, time and mass
Answer: (b)
Explanation:
Fundamental quantities must be independent. Velocity is \(\frac{\text{length}}{\text{time}}\), so it is derived and cannot be fundamental with length and time.
2. 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
(a) n ∝ size of u
(b) n ∝ u²
(c) n ∝ √u
(d) n ∝ 1/u
Answer: (d)
Explanation:
\(n_1 u_1 = n_2 u_2\), hence \(n \propto \frac{1}{u}\).
3. Suppose a quantity x can be dimensionally represented in terms of M, L and T, that is, (x) = Ma Lb Tc. The quantity mass
(a) can always be dimensionally represented in terms of L, T and x,
(b) can never be dimensoinally represented in terms of L, T and x,
(c) may be represented in terms of L, T and x if a = 0,
(d) may be represented in terms of L, T and x if a ≠ 0.
Answer: (d)
Explanation:
From \([x] = M^a L^b T^c\), mass can be expressed only if \(a \ne 0\).
4. A dimensionless quantity
(a) never has a unit
(b) always has a unit
(c) may have a unit
(d) does not exist
Answer: (c)
Explanation:
Some dimensionless quantities have units like angle in radians.
5. A unitless quantity
(a) never has a nonzero dimension
(b) always has a nonzero dimension
(c) may have a nonzero dimension
(d) does not exist
Answer: (a)
Explanation:
Unitless quantity has no dimensions, i.e., \(M^0 L^0 T^0\).
6. ∫ dx / √(2ax – x²) = an sin-1 (x/a – 1). The value of n is
(a) 0
(b) -1
(c) 1
(d) none of these.
Answer: (a)
Explanation:
LHS has dimension \(\frac{L}{\sqrt{L^2}} = L^0\). RHS is dimensionless. Hence \(a^n\) must be dimensionless, so \(n = 0\).
OBJECTIVE II
1. The dimensions ML-1 T-2 may correspond to
(a) work done by a force
(b) linear momentum
(c) pressure
(d) energy per unit volume.
Answer: (c), (d)
Explanation:
Pressure = \(\frac{\text{Force}}{\text{Area}} = \frac{MLT^{-2}}{L^2} = ML^{-1}T^{-2}\).
Energy per unit volume = \(\frac{ML^2T^{-2}}{L^3} = ML^{-1}T^{-2}\).
2. Choose the correct statement(s) :
(a) A dimensionally correct equation may be correct.
(b) A dimensionally correct equation may be incorrect.
(c) A dimensionally incorrect equation may be correct.
(d) A dimensionally incorrect equation may be incorrect.
Answer: (a), (b), (d)
Explanation:
A dimensionally correct equation may still be incorrect due to missing constants like \((2\pi)\). A dimensionally incorrect equation is always wrong.
3. Choose the correct statement(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 dimension 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.
Answer: (a), (b), (c)
Explanation:
Base quantities are independent, so one cannot be expressed in terms of others. Derived quantities can have zero power in some base quantities, e.g., velocity has zero power of mass.
EXERCISES
1. Find the dimensions of (a) linear momentum, (b) frequency and (c) pressure.
(a) \(MLT^{-1}\) (b) \(T^{-1}\) (c) \(ML^{-1}T^{-2}\)
Explanation:
Linear momentum = \(M \times LT^{-1} = MLT^{-1}\).
Frequency = \(\frac{1}{T} = T^{-1}\).
Pressure = \(\frac{MLT^{-2}}{L^2} = ML^{-1}T^{-2}\).
2. Find the dimensions of (a) angular speed \(\omega\), (b) angular acceleration \(\alpha\), (c) torque \(\Gamma\) and (d) moment of inertia \(I\).
(a) \(T^{-1}\) (b) \(T^{-2}\) (c) \(ML^2T^{-2}\) (d) \(ML^2\)
Explanation:
Angular speed = \(\frac{\text{angle}}{\text{time}} = T^{-1}\).
Angular acceleration = \(\frac{T^{-1}}{T} = T^{-2}\).
Torque = \(MLT^{-2} \times L = ML^2T^{-2}\).
Moment of inertia = \(M \times L^2 = ML^2\).
3. Find the dimensions of (a) electric field E, (b) magnetic field B and (c) magnetic permeability \(\mu_0\).
(a) \(MLT^{-3}I^{-1}\) (b) \(MT^{-2}I^{-1}\) (c) \(MLT^{-2}I^{-2}\)
Explanation:
Electric field \(E = \frac{F}{q} = \frac{MLT^{-2}}{IT} = MLT^{-3}I^{-1}\).
Magnetic field \(B = \frac{F}{qv} = \frac{MLT^{-2}}{IT \cdot LT^{-1}} = MT^{-2}I^{-1}\).
\(\mu_0 = \frac{2\pi a B}{I} = MLT^{-2}I^{-2}\).
4. Find the dimensions of (a) electric dipole moment p and (b) magnetic dipole moment M.
(a) \(LTI\) (b) \(L^2I\)
Explanation:
Electric dipole moment = \(IT \times L = LTI\).
Magnetic dipole moment = \(I \times L^2 = L^2I\).
5. Find the dimensions of Planck’s constant h from the equation \(E = h\nu\).
\(ML^2T^{-1}\)
Explanation:
\(h = \frac{E}{\nu} = \frac{ML^2T^{-2}}{T^{-1}} = ML^2T^{-1}\).
6. Find the dimensions of (a) the specific heat capacity c, (b) the coefficient of linear expansion α and (c) the gas constant R.
(a) \(L^2 T^{-2} K^{-1}\) (b) \(K^{-1}\) (c) \(ML^2 T^{-2} K^{-1} (mol)^{-1}\)
Explanation:
\(c = \frac{Q}{m(T_2 – T_1)} = \frac{ML^2T^{-2}}{M \cdot K} = L^2T^{-2}K^{-1}\).
\(\alpha = \frac{\Delta l}{l_0 \Delta T} = \frac{L}{L \cdot K} = K^{-1}\).
\(R = \frac{PV}{nT} = \frac{(ML^{-1}T^{-2}) \cdot L^3}{mol \cdot K} = ML^2T^{-2}K^{-1}(mol)^{-1}\).
7. Taking force, length and time to be the fundamental quantities find the dimensions of (a) density, (b) pressure, (c) momentum and (d) energy.
(a) \(FL^{-4}T^2\) (b) \(FL^{-2}\) (c) \(FT\) (d) \(FL\)
Explanation:
Mass = \(FL^{-1}T^2\).
Density = \(\frac{M}{L^3} = \frac{FL^{-1}T^2}{L^3} = FL^{-4}T^2\).
Pressure = \(\frac{F}{L^2} = FL^{-2}\).
Momentum = \(F \cdot T = FT\).
Energy = \(F \cdot L = FL\).
8. Suppose the acceleration due to gravity at a place is 10 m/s². Find its value in cm/(minute)² .
\(36 \times 10^5\ \text{cm/(minute)}^2\)
Explanation:
\(10\ \text{m/s}^2 = 10 \times \frac{100\ \text{cm}}{(1/60\ \text{min})^2} = 10 \times 100 \times 3600 = 36 \times 10^5\ \text{cm/(minute)}^2\).
9. 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.
\(0.0089\ \text{m/s},\ 31\ \text{m/s}\)
Explanation:
\(1\ \text{mile} = 1609\ \text{m},\ 1\ \text{hour} = 3600\ \text{s}\).
Snail: \(0.020 \times \frac{1609}{3600} \approx 0.0089\ \text{m/s}\).
Leopard: \(70 \times \frac{1609}{3600} \approx 31\ \text{m/s}\).
10. 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 = 13.6, Density of water = 10³ kg/m³, g = 9.8 m/s² at Calcutta. Pressure = hρg in usual symbols.
\(10 \times 10^4\ \text{N/m}^2,\ 10 \times 10^5\ \text{dyne/cm}^2\)
Explanation:
\(\rho = 13.6 \times 10^3\ \text{kg/m}^3\).
\(P = 0.75 \times 13.6 \times 10^3 \times 9.8 \approx 10 \times 10^4\ \text{N/m}^2\).
\(1\ \text{N/m}^2 = 10\ \text{dyne/cm}^2\), so \(10 \times 10^5\ \text{dyne/cm}^2\).
11. Express the power of a 100 watt bulb in CGS unit.
\(10^9\ \text{erg/s}\)
Explanation:
\(1\ \text{W} = 1\ \text{J/s},\ 1\ \text{J} = 10^7\ \text{erg}\).
\(100\ \text{W} = 100 \times 10^7 = 10^9\ \text{erg/s}\).
12. The normal duration of I.Sc. Physics practical period in Indian colleges is 100 minutes. Express this period in microcenturies. 1 microcentury = \(10^{-6} \times 100\) years. How many microcenturies did you sleep yesterday ?
\(1.9\ \text{microcenturies}\)
Explanation:
\(1\ \text{microcentury} = 10^{-6} \times 100 \times 365 \times 24 \times 60 \approx 52.56\ \text{minutes}\).
\(100 / 52.56 \approx 1.9\ \text{microcenturies}\).
13. The surface tension of water is 72 dyne/cm. Convert it in SI unit.
\(0.072\ \text{N/m}\)
Explanation:
\(1\ \text{dyne} = 10^{-5}\ \text{N},\ 1\ \text{cm} = 10^{-2}\ \text{m}\).
\(72 \times \frac{10^{-5}}{10^{-2}} = 72 \times 10^{-3} = 0.072\ \text{N/m}\).
14. The kinetic energy K of a rotating body depends on its moment of inertia I and its angular speed \(\omega\). Assuming the relation to be \(K = kI^a \omega^b\) where k is a dimensionless constant, find a and b.
\(a = 1,\ b = 2\)
Explanation:
\(K = ML^2T^{-2},\ I = ML^2,\ \omega = T^{-1}\).
\(ML^2T^{-2} = (ML^2)^a (T^{-1})^b = M^a L^{2a} T^{-b}\).
Comparing powers: \(a = 1,\ b = 2\).
15. 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.
\(E = kmc^2\)
Explanation:
Let \(E = km^a c^b\).
\(ML^2T^{-2} = M^a (LT^{-1})^b = M^a L^b T^{-b}\).
Comparing powers: \(a = 1,\ b = 2\).
Hence \(E = kmc^2\).
16. 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.
\(V = IR\)
Explanation:
\(I \times R = I \times (ML^2 I^{-2} T^{-3}) = ML^2 I^{-1} T^{-3}\), which matches dimensions of \(V\).
17. The frequency of vibration of a string depends on the length L, the tension F and its mass per unit length m. Guess the expression for its frequency from dimensional analysis.
\(\nu = \frac{k}{L} \sqrt{\frac{F}{m}}\)
Explanation:
\(\nu = kL^a F^b m^c\).
\(T^{-1} = L^a (MLT^{-2})^b (ML^{-1})^c\).
Solving: \(b=\frac{1}{2},\ c=-\frac{1}{2},\ a=-1\).
18. Test if the following equations are dimensionally correct :
(a) \(h = \frac{2S \cos\theta}{\rho r g}\),
(b) \(v = \sqrt{\frac{P}{\rho}}\),
(c) \(V = \frac{\pi P r^4 t}{8 \eta l}\),
(d) \(\nu = \frac{1}{2\pi} \sqrt{\frac{mgl}{I}}\).
Answer: all are dimensionally correct
Explanation:
Each equation satisfies dimensional homogeneity: (a) \(L\), (b) \(LT^{-1}\), (c) \(L^3\), (d) \(T^{-1}\).
19. Let x and a stand for distance. Is \(\int \frac{dx}{\sqrt{a^2 – x^2}} = \frac{1}{a} \sin^{-1} \left(\frac{a}{x}\right)\) dimensionally correct ?
No
Explanation:
LHS is dimensionless \((L/L)\). RHS has factor \(1/a = L^{-1}\), so dimensions do not match.