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.
Constants which possess dimensions are called dimensional constants.
Example: Planck’ Constant.
Those physical quantities which possess dimensions but do not have a fixed value are called dimensional variables.
Example: Displacement, Force, velocity etc.
Physical quantities which do not possess dimensions are called dimensionless quantities.
Example: Angle, specific gravity, strain. In general, physical quantity which is a ratio of two quantities of same dimension will be dimensionless.
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.
in equation X + Y = Z is valid if the dimensions of X, Y and Z are same.
The applications of dimensional analysis are:
Limitations of Dimensional Analysis are:
We know that that the units that depend upon the fundamental units of mass, length and time are called derived units. The unit of mass, length and time are denoted by M, I and T. ( The dimensions of a derived unit may be defined as the powers to which the fundamental units of mass, length and time must be raised so as to completely represent it.
It is an compound expression, showing how and which of the fundamental units enter into the unit of a physical quantity.
It is an expression which expresses the physical quantity in terms of a fundamental units of mass, length and time.
Physical quantities which are variable but have no dimensions are called non - dimensional variable,
Example: strain, specific gravity, angle etc.
It is a term used for lumber that is finished and cut to standerdized width and depth specified in inches.
According to this principle, the dimensions of all the terms on the two sides of an equation must be same.Therefore in a given relation the terms on either side have same dimensions, If the relation is a correct one, but if it is not so, th erelation is not correct
The error which creeps in during a measurement due to individual measuring person and the care taken by him in the measuring process is called random error. In order to minimise this error, measurements are repeated many times.
The error which creeps in during a measurement due to limit or resolution of the measuring instrument is called instrumental error.
The magnitude of the difference between the true value ( i.e. the mean )of the quantity and the individual measured value is called absolute error.
It indicates the extent to which the reading are reliable.
There are only seven base units and two supplementary units.
The word kinematics is derived from the greek word “ Knemia ” which means motion. Thus kinematics is the study of motion. We study the position, velocity, acceleration etc. of a body without specifyng the nature of the body and the nature of the forces which cause motion. In this branch we study ways to describe motion of object independent of causes of motion and independent of the nature of the body.
It is dervied from the greek word “ dynamics ” which means power. It deals with the study of motion taking into consideration the forces which cause motion.
It is the study of objects at rest i.e. when a large number of forces acting on a body are in equilibrium.
This deals with all the subjects namely, kinematics, dynamics and statics.
It is the change of position of an object in the course of time.
The body in motion is treated as a particle.
The motion has been classfied as :-
Both are motions in two dimensions
If we assign a negative time to an event, it means that it occured before the event to which positive time was assigned.
The position co - ordinate in a moving body describes te definite and exact position of the body at any time. The position of a body at any instant is called instantaneous position.
A motion is said to be uniform if the body moves equal distances in equal intervals of time and always in the same direction. For such a motion, the actual distance covered in time t is the magnitude of the displacement.
If the body moves equal distances in equal intervals of time and always in the same direction, then it is said to possess uniform velocity.
When a body travels unequal distance in equal intervals of time, the motion is said to be non - uniform motion.
If a body covers unequal distances in equal intervals of time along a straight line or if the body changes the direction of motion ( though it may be covering equal intervals of time ) , it is said to process variable velocity.
It is the ratio of the total distance travelled to the total time taken by the body.
The velocity of a body in a non - uniform motion at any instant is called instantaneous velocity. It is different from the average velocity over an interval of time.
It is the rate of change of velocity with time.
A body is said to be moving with uniform acceleration, if its velocity changes by equal values in equal intervals of time.
If the motion changes of a body is such that its velocity changes by unequal values in equal intervals of time, then the value of the accleration at any instant is called instantaneous acceleration.
If the velocity increases, the acceleration is positive and if the velocity decreases, the acceleration is negative. The negative acceleration is called retardation.
Velocity is vertically upward and acceleration is vertically downwards.
Velocity is vertically downward and acceleration is also vertically downwards.
With the help of this graph, we can determine, distance travelled during any interval of time and also the velocity of the body at any instant of time.
It is found that the basic laws of motion involve only acceleration and not the rate of change of acceleration, so we never consider the rate of change of acceleration.
The angle that the of projection makes with the horizontal is called angle of departure or angle of projection. Clearly angle of projection for a horizontal projectile is zero.
The distance between the point of projection and the point where the trajectory meets the horizontal plane through the point of projection is called its range ( horizontal ).
The horizontal component of the velocity of the body remains same throughout because there is no acceleration ( due to gravity ) in the horizontal direction.
The vertical component of the veocity initially ( i.e. at t = 0 ) is zero and the vertical component keeps on increasing till body touches the ground.
Velocity is the rate of change of the position, equal to speed in a particular direction.
Satyendranath Bose, who with Einstein developed a system of statical quantum mecahnics now known sa Bose Einstein Statistics.
The minimum speed that a space rocket must reach to escape the earth’s gravity.
The basic forces are gravity, electricity, magnetism and two kinds of nuclear forces called weak and strong forces.
Abdus Salam became the first person from pakistan who won a nobel prize for prove this theory.
Some forces are only produced when the one object touches another. These force are called non - contact forces.
This is a state of a system in which it is apparently in a stable equilibrium, however if slightly distrubed the system changes to a new state of lower energy.
A method used to find a relation between various physical quantities. Also to calculate how a physical quantity will depend in terms of the powers of fundamental units on which it intuitively depends.
The method is based on the prinicple that the dimensions of the fudamental quantities ( M, L and T ) must be the same on both sides of an equation.
These are the powers to which the fundamental units must be raised, when the quantity is expressed interms of these units.
Question 55. (ncert): A Book With Many Printing Errors Contains Four Different Formulas For The Displacement 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.
Question 56. (ncert): A Famous Relation In Physics Relates ‘moving Mass’ M To The ‘rest Mass’ Mo Of A Particle In Terms Of Its Speed V And The Speed Of Light, C. (this Relation First Arose As A Consequence Of Special Relativity Due To Albert Einstein). A Boy Recalls The Relation Almost Correctly But Forgets Where To Put The Constant C. He Writes?
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’
Question 57. 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]
Question 58. 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?
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
Question 59. 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?
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] …..
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]
Question 61. 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?
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
Question 62. 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
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
Question 64. 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)?
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+cT-b]
⇒ a =1, b +c = 1, -b = -2
⇒ a= 1, b = 2, c = -1
∴ F = km1.v2.r-1= kmv2/r
Question 65. 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 108dyn cm2/gm2 …(II)
Dimensions of h = [M1L2T-1] = 6.6 x 10-27erg 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 10-27erg sec)/(3 x 1010 cm/sec)3
⇒ [L2] = 1.6304 x 10-65cm2
⇒ [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 1010cm/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 10-42s = 7.42 x 1042unit of time
Question 66. 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?
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
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]
Physical Quantities having the same dimensional formula:
a. impulse and momentum.
b. force, thrust.
c. work, energy, torque, moment of force, energy
d. angular momentum, Planck’s constant, rotational impulse
e. force constant, surface tension, surface energy.
f. stress, pressure, modulus of elasticity.
g. angular velocity, frequency, velocity gradient
h. latent heat, gravitational potential.
i. thermal capacity, entropy, universal gas constant and Boltzmann’s constant.
j. power, luminous flux.
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
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