NCERT Chapter Solutions and Other Numerical Problems
|Joseph Fourier - |
How is he related to Dimensions?
Answer: The term 'dimension' is used to refer to the physical nature of a quantity and the type of unit used to specify it. Mathematically dimensions of a physical quantity are the powers to which the fundamental quantities must be raised.
e.g. Dimension of velocity = Displacement / time = [L]/[T] = [M0][L1][T-1]
Q2: What are dimensional constants?
Answer: Constants which possess dimensions are called dimensional constants. E.g. Planck' Constant.
Q3: What are dimensional variables?
Answer: Those physical quantities which possess dimensions but do not have a fixed value are called dimensional variables. E.g. Displacement, Force, velocity etc.
Q4: What are dimensionless quantities?
Answer: Physical quantities which do not possess dimensions are called dimensionless quantities. E.g. Angle, specific gravity, strain. In general, physical quantity which is a ratio of two quantities of same dimension will be dimensionless.
Q5: Define the principle of homogeneity of dimensions. On What principle is it based?
Answer: The principle of homogeneity of dimensions states that an equation is dimensionally correct if the dimensions of the various terms on either side of the equation are the same.
This principle is based on the fact that two quantities of the same dimension only can be added up, and the resulting quantity also possess the the same dimension.
i.e. In equation X + Y = Z is valid if the dimensions of X, Y and Z are same.
Q6: Who introduced Dimension Analysis
Answer: Fourier (Joseph Fourier - French Mathematician)
Q7: List the basic dimensions.
- Length - L
- Time - T
- Mass - M
- Temperature - K or θ
- Current - A
Answer: The applications of dimensional analysis are:
- To convert a physical quantity from one system of units to another.
- To check the dimensional correctness of a given equation.
- establish a relationship between different physical quantities in an equation.
y of a particle undergoing a certain periodic motion:
(a) y = a sin 2π t/T
(b) y = a sin vt
(c) y = (a/T) sin t/a
(d) y = (a 2) (sin 2πt / T + cos 2πt / T )
(a = maximum displacement of the particle, v = speed of the particle. T = time-period
of motion). Rule out the wrong formulas on dimensional grounds.
Dimension of a = displacement = [M0L1T0]
Dimension of v (speed) = distance/time = [M0L1T-1]
Dimension of t or T (time period) = [M0L0T1]
Trigonometric function sine is a ratio, hence it must be dimensionless.
(a) y = a sin 2π t/T (correct ✓ )
Dimensions of RHS = [L1] sin([T].[T-1] ) = [M0L1T0] = LHS (eqation is correct).
(b) y = a sin vt (wrong ✗)
RHS = [L1] sin([LT-1] [T1]) = [L1] sin([L]) = wrong, since trigonometric function must be dimension less.
(c) y = (a/T) sin t/a (wrong ✗)
RHS = [L1] sin([T].[L-1] ) = [L1] sin([TL-1] ) = wrong, sine function must be dimensionless.
(d) y = (a 2) (sin 2πt / T + cos 2πt / T ) (correct ✓ )
RHS = [L1] ( sin([T].[T-1] + cos([T].[T-1] ) = [L1] ( sin(M0L1T0) + cos(M0L1T0) )
= [L1] = RHS = equation is dimensionally correct.
Dimension of m (mass) = [M1L0T0]
Dimension of m0 (mass) = [M1L0T0]
Dimension of v (velocity) = [M0L1T-1]
∴ Dimension of v2 = [M0L2T-2]
Dimension of c (velocity) = [M0L1T-1]
Applying principle of homogeneity of dimensions, [LHS] = [RHS] = [M1L0T0]
⇒ The equation (1- v2)½ must be dimension less, which is possible if we have the expressions as:
(1 - v2/c2) The equation after placing 'c' will be:
Q11: Check the following equation for calculating displacement is dimensionally correct or not
(a) x = x0 + ut + (1/2) at2
where, x is displacement at given time t
xo is the displacement at t = 0
u is the velocity at t = 0
a represents the acceleration.
(b) P = (ρgh)½
where P is the pressure,
ρ is the density
g is gravitational acceleration
h is the height.
(a) x = x0 + ut + (1/2) at2
Applying principle of homogeneity, all the sub-expressions of the equation must have the same dimension and be equal to [LHS]
Dimension of x = [M0L1T0]
Dimensions of sub-expressions of [RHS] must be [M0L1T0]
⇒ Dimension of x0 (displacement) = [M0L1T0] = [LHS]
Dimension of ut = velocity x time = [M0L1T-1][M0L0T1] = [M0L1T0] = [LHS]
Dimension of at2 = acceleration x (time)2 = [M0L1T-2][M0L0T-2] = [M0L1T0] = [LHS]
∴ The equation is dimensionally correct.
(b) P = (ρgh)½
Dimensions of LHS i.e. Pressure [P] = [M1L-1T-2]
Dimensions of ρ = mass/volume = [M1L-3T0]
Dimensions of g (acceleration) = [M0L1T-2]
Dimensions of h (height) = [M0L1T0]
Dimensions of RHS = [(ρgh)½] = ([M1L-3T0]. [M0L1T-2].[M0L1T0])½ = ([M1L-1T-2])½
= [M½L-½T-1] ≠ [LHS]
The given equation is not dimensionally correct.
Q12: Prove that trigonometric function like sin θ are dimensionless.
Answer: As shown in figure, sin θ = y/r. = [L]/[L] = 1
⇒ sin θ is dimensionless
Q13 (NCERT): A man walking briskly in rain with speed v must slant his umbrella forward making an angle θ with the vertical. A student derives the following relation between θ and v : tan θ = v and checks that the relation has a correct limit: as v → 0, θ →0, as expected. (We are assuming there is no strong wind and that the rain falls vertically for a stationary man). Do you think this relation can be correct ? If not, guess the correct relation.
Answer: Given, v = tanθ
Dimensions of LHS = [v] = [M0L1T-1]
Dimension of RHS = [tanθ] = [M0L0T0] (trigonometric ratios are dimensionless)
Since [LHS] ≠ [RHS]. Equation is dimensionally incorrect.
To make the equation dimensionally correct, LHS should also be dimension less. It may be possible if consider speed of rainfall (Vr) and the equation will become:
tan θ = v/Vr
Q14: Hooke’s law states that the force, F, in a spring extended by a length x is given by F = −kx.
According to Newton’s second law F = ma, where m is the mass and a is the acceleration.
Calculate the dimension of the spring constant k.
Answer: Given, F = -kx
⇒ k = - F/x
F = ma, the dimensions of force is:
[F] = ma = [M1L0T0].[M0L1T-2] = [M1L1T-2]
Therefore, dimension of spring constant (k) is:
[k] = [F]/[x] = [M1L1T-2].[M0L-1T0] = [M1L0T-2] or [MT-2] ..... (answer)
Q15: Compute the dimensional formula of electrical resistance (R).
Answer: According to Ohm's law
V = IR or R = V/I
Since Work done (W) = QV where Q is the charge
⇒ R = W/QI = W/I2t (I = Q/t)
Dimensions of Work [W] = [M1L2T-2]
∴Dimension of R = [R] = [M1L2T-2][A-2T-1] = [M1L2T-3A-2] ... (answer)
Q16: Convert 76 cm of mercury pressure into N m-2 using the dimensional formula. (Density of Hg = 13.6 gm/mL)
Pressure P = hdg (h = height, d = density, g = acceleration due to gravity).
In cgs system, 76 cm of Hg will exert pressure (P1), i.e.
P1 = 76 x 13.6 x 980 = 1012928 dynes cm-2.
Dimension of Pressure in [M1L-1T-2]
Applying to the formula (shown above in note), a = 1, b = -1 and c = -2
Q17: A calorie is a unit of heat or energy and it equals about 4.2 J where 1J = 1 kg m2 s–2. Suppose we employ a system of units in which the unit of mass equals α kg, the unit of length equals β m, the unit of time is γ s. Show that a calorie has a magnitude 4.2 α–1 β–2 γ2 in terms of the new units.
Answer: Considering the unit conversion formula,
n1U1 = n1U2
n1[M1aL1bT1c] = n2[M2aL2bT2c]
Given here, 1 Cal = 4.2 J = 4.2 kg m2 s–2.
n1 = 4.2, M1 = 1kg, L1 = 1m, T1 = 1 sec
n2 = ?, M2 = α kg, L2 = βm, T2 = γ sec
The dimensional formula of energy is = [M1L2T-2]
⇒ a = 1, b =1 and c = -2 Putting these values in above equation,
= 4.2[1Kg/α kg]1[1m/βm]2[1sec/γ sec]-2 = 4.2 α–1 β–2 γ2 ... (answer)
Q18: The kinetic energy K of a rotating body depends on its moment of inertia I and its angular speed ω. Considering the relation to be K = kIaωb where k is dimensionless constant.
Find a and b. Moment of Inertia of a spehere about its diameter is (2/5)Mr2
Dimensions of Kinetic energy K = [M1L2T-2]
Dimensions of Moment of Inertia (I) = [ (2/5)Mr2] = [ML2T0]
Dimensions of angular speed ω = [θ/t] = [M0L0T-1]
Applying principle of homogeneity in dimensions in the equation K = kIaωb
[M1L2T-2] = k ( [ML2T0])a([M0L0T-1])b
[M1L2T-2] = k [MaL2aT-b]
⇒ a = 1 and b = 2
⇒ K = kIω2 ... (answer)
Q19: What are the limitations of Dimensional Analysis?
Answer: Limitations of Dimensional Analysis are:
- It cannot determine value of dimensionless constants.
- We cannot use this method to equations involving exponential and trigonometric functions.
- It cannot be applied to an equation involving more than three physical quantities.
- It is a too not a solution i.e. It can check only if the equation is dimensionally correct or not. But cannot say the equation is absolutely correct.
Q20: Convert 1 Newton into dyne using method of dimensions.
Answer: Dimensions of Force = [M1L1T-2]
Considering dimensional unit conversion formula i.e. n1[M1aL1bT1c] = n2[M2aL2bT2c]
⇒ a = 1, b = 1 and c = -2
In SI system, M1 = 1kg, L1 = 1m and T1 = 1s
In cgs system, M2 = 1g, L2 = 1cm and T2 = 1s
Putting the values in the conversion formula,
n2 = n1(1Kg/1g)1.(1m/1cm)1(1s/1s)-2 = 1.(103/1g)(102cm) = 105dyne ...(answer)
Q21: The centripetal force (F) acting on a particle (moving uniformly in a circle) depends on the mass (m) of the particle, its velocity (v) and radius (r) of the circle. Derive dimensionally formula for force (F).
Answer: Given, F ∝ ma.vb.rc
∴ F = kma.vb.rc (where k is constant)
Putting dimensions of each quantity in the equation,
[M1L1T-2] = [M1L0T0]a. [M0L1T-1]b. [M0L1T0]c = [MaLb+cT-b]
⇒ a =1, b +c = 1, -b = -2
⇒ a= 1, b = 2, c = -1
∴ F = km1.v2.r-1 = kmv2/r
Q22: If the velocity of light c, gravitational constant G and planks constant h be chosen as fundamental units, find the value of a gram, a cm and a sec in term of new unit of mass, length and time respectively.
(Take c = 3 x 1010 cm/sec, G = 6.67 x 108 dyn cm2/gram2 and h = 6.6 x 10-27 erg sec)
c = 3 x 1010 cm/sec
G = 6.67 x 108 dyn cm2/gm2
h = 6.6 x 10-27 erg sec
Putting respective dimensions,
Dimension formula for c = [M0L1T-1] = 3 x 1010 cm/sec .... (I)
Dimensions of G = [M-1L3T-2] = 6.67 x 108 dyn cm2/gm2 ...(II)
Dimensions of h = [M1L2T-1] = 6.6 x 10-27 erg sec ...(III)
(Note: Applying newton's law of gravitation, you can find dimensions of G i.e. G = Fr2/(mM)
Similarly, Planck's Constant (h) = Energy / frequency)
To get M, multiply eqn-I and III and divide by eqn.-II,
= ( 3 x 1010 cm/sec).( 6.6 x 10-27 erg sec)/ 6.67 x 108 dyn cm2/gm2
⇒[M2] = 2.968 x 10-9
⇒[M] = 0.5448 x 10-4 gm
or 1gm = [M]/0.5448 x 10-4 = 1.835 x 10-4 unit of mass
To obtain length [L], eqn.-II x eqn-III / cube of eqn.-I i.e.
= (6.67 x 108 dyn cm2/gm2 ).( 6.6 x 10-27 erg sec)/(3 x 1010 cm/sec)3
⇒ [L2] = 1.6304 x 10-65 cm2
⇒ [L] = 0.4038 x 10-32 cm
or 1cm = [L]/ 0.4038 x 10-32 = 2.47 x 10-32unit of length
In eqn-I, [M0L1T-1] = 3 x 1010 cm/sec
⇒ [T] = [L] ÷ 3 x 1010cm/s
⇒ [T] = 0.4038 x 10-32 cm ÷ 3 x 1010cm/s = 0.1345 x 10-42 s
or 1s = [T]/0.1345 x 10-42 s = 7.42 x 1042 unit of time
Q 23: A student while doing an experiment finds that the velocity of an object varies with time and it can be expressed as equation:
v = Xt2 + Yt +Z .
If units of v and t are expressed in terms of SI units, determine the units of constants X, Y and Z in the given equation.
Answer: Given, v = Xt2 + Yt +Z
Dimensions of velocity v = [M0L1T-1]
Applying applying principle of homogeneity in dimensions, terms must have same dimension.
[v] = [Xt2] + [Yt] + [Z]
∴ [v] = [Xt2]
⇒ [X] = [v] /[t2] = [M0L1T-1] / [M0L0T2] = [M0L1T-3] ....(i)
Similarly, [v] = [Yt]
⇒ [Y] = [v] / [t] = [M0L1T-1]/ [M0L0T-1] = [M0L1T-2] ...(ii)
Similarly, [v]= [Z]
[Z] = [M0L1T-1] ...(iii)
⇒ Unit of X = m-s-3
⇒ Unit of Y = m-s-2
⇒ Unit of Z = m-s-1
Q24: Express Capacitance in terms of dimensions of fundamental quantities i.e. Mass (M), Length(L), Time(T) and Ampere(A)
Answer: Capacitance(C) is defined as the ability of a electric body to store electric charge.
∴ Capacitance (C) = Total Charge(q) / potential difference between two plates (V)
= Coulomb/ Volt
∵ Volt = Work done (W)/ Charge(q) = Joule/Coulomb
⇒ Capacitance (C) = Charge(q)2/ Work(W)
∵ Charge (q) = Current (I) × Time(t)
Dimension of [q] = [AT] ----------- (I)
Dimension of Work = Force × distance = [MLT-2][L] = [ML2T-2] --------- (II)
Putting values of I and II,
[C] = ([AT])2/ [ML2T-2] = [M-1L-2T2+2A2] = [M-1L-2T4A2]
Q25: If Force (F), velocity (V) and acceleration (A) are taken as the fundamental units instead of mass, length and time, express pressure and impulse in terms of F, V and A.
Answer: We know that Force = mass ✕ acceleration
⇒ mass = FA-1
and length = velocity ✕ time = velocity ✕ velocity ÷ acceleration = V2A-1
and time = VA-1
∵ Pressure = Force ÷ Area = F ÷ (V2A-1)2 = FV-4A2
Impulse = Force ✕ time = FVA-1