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Question 1. Define The Term ‘dimension’?
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.
Question 2. What Are Dimensional Constants?
Answer :
Constants which possess dimensions are called dimensional constants.
Example: Planck’ Constant.
Question 3. What Are Dimensional Variables?
Answer :
Those physical quantities which possess dimensions but do not have a fixed value are called dimensional variables.
Example: Displacement, Force, velocity etc.
Question 4. What Are Dimensionless Quantities?
Answer :
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.
Question 5. 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.
in equation X + Y = Z is valid if the dimensions of X, Y and Z are same.
Question 6. List The Basic Dimensions?
Answer :
Question 7. What Are The Uses (applications) Of Dimensional Analysis?
Answer :
The applications of dimensional analysis are:
Question 8. What Are The Limitations Of Dimensional Analysis?
Answer :
Limitations of Dimensional Analysis are:
Question 9. What Do You Mean By “ Dimensions Of A Derived Unit ” ?
Answer :
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.
Question 10. What Id Dimensional Formula ?
Answer :
It is an compound expression, showing how and which of the fundamental units enter into the unit of a physical quantity.
Question 11. What Is Dimensional Equation?
Answer :
It is an expression which expresses the physical quantity in terms of a fundamental units of mass, length and time.
Question 12. What Are Non- Dimensional Variable?
Answer :
Physical quantities which are variable but have no dimensions are called non - dimensional variable,
Example: strain, specific gravity, angle etc.
Question 13. What Is Dimensional Lumber?
Answer :
It is a term used for lumber that is finished and cut to standerdized width and depth specified in inches.
Question 14. What Is The Principle Of Homogeneity Of Dimensons ?
Answer :
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
Question 15. What Is Random Error?
Answer :
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.
Question 16. What Is Meant By Instrumental Error?
Answer :
The error which creeps in during a measurement due to limit or resolution of the measuring instrument is called instrumental error.
Question 17. What Is Meant By Absolute Error In A Measurement?
Answer :
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.
Question 18. What Is Meant By Significant Figures ?
Answer :
It indicates the extent to which the reading are reliable.
Question 19. How Many Base Units Are There In The S.i System ?
Answer :
There are only seven base units and two supplementary units.
Question 20. What Is Kinematics?
Answer :
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.
Question 21. What Is Dynamics?
Answer :
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.
Answer :
It is the study of objects at rest i.e. when a large number of forces acting on a body are in equilibrium.
Question 23. What Is Mechanics?
Answer :
This deals with all the subjects namely, kinematics, dynamics and statics.
Answer :
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 :-
Question 25. Which Type Of Motions Are The Following ?
Answer :
Both are motions in two dimensions
Question 26. What Is Meant By Negative Time And Positive Time?
Answer :
If we assign a negative time to an event, it means that it occured before the event to which positive time was assigned.
Question 27. What Is Meant By Instanteous Position Of An Object?
Answer :
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.
Question 28. What Is Uniform Motion?
Answer :
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.
Question 29. What Is Uniform Velocity?
Answer :
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.
Question 30. What Is Non - Uniform Motion?
Answer :
When a body travels unequal distance in equal intervals of time, the motion is said to be non - uniform motion.
Question 31. What Is Variable Velocity?
Answer :
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.
Question 32. What Is Average Velocity?
Answer :
It is the ratio of the total distance travelled to the total time taken by the body.
Question 33. What Is Instantaneous Velocity?
Answer :
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.
Question 34. What Is Acceleration?
Answer :
It is the rate of change of velocity with time.
Question 35. What Is Uniform Acceleration?
Answer :
A body is said to be moving with uniform acceleration, if its velocity changes by equal values in equal intervals of time.
Question 36. What Is Instantaneous Acceleration?
Answer :
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.
Question 37. What Is Retardation?
Answer :
If the velocity increases, the acceleration is positive and if the velocity decreases, the acceleration is negative. The negative acceleration is called retardation.
Question 38. What Is The Direction Of The Velocity And Acceleration When It Is Thrown Upwards ?
Answer :
Velocity is vertically upward and acceleration is vertically downwards.
Question 39. What Is The Direction Of The Velocity And Acceleration When It Is Thrown Down Wards?
Answer :
Velocity is vertically downward and acceleration is also vertically downwards.
Question 40. How Is The Position Time Graph Helpful In Studying The Motion Of The Body?
Answer :
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.
Question 41. Why We Do Not Ever Consider Rate Of Change Of Acceleration?
Answer :
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.
Question 42. What Is Angle Of Departure Or Angle Of Projection?
Answer :
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.
Question 43. What Is Range Of Projectile?
Answer :
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 ).
Question 44. What Is Horizontal Velocity Of Projectile ?
Answer :
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.
Question 45. What Is Velocity?
Answer :
Velocity is the rate of change of the position, equal to speed in a particular direction.
Question 46. Which Indian Physcist Worked With Albert Einstein ?
Answer :
Satyendranath Bose, who with Einstein developed a system of statical quantum mecahnics now known sa Bose Einstein Statistics.
Question 47. What Is Escape Velocity?
Answer :
The minimum speed that a space rocket must reach to escape the earth’s gravity.
Question 48. Which Are The Basic Forces?
Answer :
The basic forces are gravity, electricity, magnetism and two kinds of nuclear forces called weak and strong forces.
Question 49. Which Scientist Proved The Electro Weak Force They?
Answer :
Abdus Salam became the first person from pakistan who won a nobel prize for prove this theory.
Question 50. Which Are Called Non - Contact Forces?
Answer :
Some forces are only produced when the one object touches another. These force are called non - contact forces.
Question 51. What Is Metastable State?
Answer :
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.
Question 52. What Is Dimensional Analysis?
Answer :
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.
Question 53. What Is Physical Quantity Of Dimesion?
Answer :
These are the powers to which the fundamental units must be raised, when the quantity is expressed interms of these units.
Question 54. What Are The Main Uses Of Dimensional Equations?
Answer :
Answer :
Given,
Dimension of a = displacement = [M^{0}L^{1}T^{0}]
Dimension of v (speed) = distance/time = [M^{0}L^{1}T^{-1}]
Dimension of t or T (time period) = [M^{0}L^{0}T^{1]}
Trigonometric function sine is a ratio, hence it must be dimensionless.
(a) y = a sin 2π t/T (correct ✓ )
Dimensions of RHS = [L^{1}] sin([_{T}].[T^{-1}] ) = [M^{0}L^{1}T^{0}] = LHS (eqation is correct).
(b) y = a sin vt (wrong ✗)
RHS = [L^{1}] sin([LT^{-1}] [T^{1}]) = [L^{1}] sin([L]) = wrong, since trigonometric function must be dimension less.
(c) y = (a/T) sin t/a (wrong ✗)
RHS = [L^{1}] sin([T].[L^{-1}] ) = [L^{1}] sin([TL^{-1}] ) = wrong, sine function must be dimensionless.
(d) y = (a 2) (sin 2πt / T + cos 2πt / T ) (correct ✓ )
RHS = [L^{1}] ( sin([T].[T^{-1}] + cos([T].[T^{-1}] ) = [L^{1}] ( sin(M^{0}L^{1}T^{0}) + cos(M^{0}L^{1}T^{0}) )
= [L^{1}] = RHS = equation is dimensionally correct.
Answer :
Dimension of m (mass) = [M^{1}L^{0}T^{0}]
Dimension of m0 (mass) = [M^{1}L^{0}T^{0}]
Dimension of v (velocity) = [M^{0}L^{1}T^{-1}]
∴ Dimension of v^{2}= [M^{0}L^{2}T^{-2}]
Dimension of c (velocity) = [M^{0}L^{1}T^{-1}]
Applying principle of homogeneity of dimensions, [LHS] = [RHS] = [M^{1}L^{0}T^{0}]
⇒ The equation (1- v^{2})½ must be dimension less, which is possible if we have the expressions as:
(1 – v^{2}/c^{2}) The equation after placing ‘c’
Answer :
(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 = [M^{0}L^{1}T^{0}]
Dimensions of sub-expressions of [RHS] must be [M^{0}L^{1}T^{0}]
⇒ Dimension of x0 (displacement) = [M^{0}L^{1}T^{0}] = [LHS]
Dimension of ut = velocity x time = [M^{0}L^{1}T^{-1}][M^{0}L^{0}T^{1}] = [M^{0}L^{1}T^{0}] = [LHS]
Dimension of at^{2} = acceleration x (time)2 = [M^{0}L^{1}T^{-2}][M^{0}L^{0}T^{-2}] = [M^{0}L^{1}T^{0}] = [LHS]
∴ The equation is dimensionally correct.
(b) P = (ρgh)½
Dimensions of LHS i.e. Pressure [P] = [M^{1}L^{-1}T^{-2}]
Dimensions of ρ = mass/volume = [M^{1}L^{-3}T^{0}]
Dimensions of g (acceleration) = [M^{0}L^{1}T^{-2}]
Dimensions of h (height) = [M^{0}L^{1}T^{0}]
Dimensions of RHS = [(ρgh)½] = ([M^{1}L^{-3}T^{0}]. [M^{0}L^{1}T^{-2}].[M^{0}L^{1}T^{0}])½ = ([M^{1}L^{-1}T^{-2}])½
= [M^{½}L^{-½}T^{-1}] ≠ [LHS]
Answer :
Given, v = tanθ
Dimensions of LHS = [v] = [M^{0}L^{1}T^{-1}]
Dimension of RHS = [tanθ] = [M^{0}L^{0}T^{0}] (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
Answer :
Given, F = -kx
⇒ k = – F/x
F = ma,
the dimensions of force is:
[F] = ma = [M^{1}L^{0}T^{0}].[M^{0}L^{1}T^{-2}] = [M^{1}L^{1}T^{-2}]
Therefore, dimension of spring constant (k) is:
[k] = [F]/[x] = [M^{1}L^{1}T^{-2}].[M^{0}L^{-1}T^{0}] = [M^{1}L^{0}T^{-2}] or [MT^{-2}] …..
Question 60. 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/I^{2}t (I = Q/t)
Dimensions of Work [W] = [M^{1}L^{2}T^{-2}]
∴Dimension of R = [R] = [M^{1}L^{2}T^{-2}][A^{-2}T^{-1}] = [M^{1}L^{2}T^{-3}A^{-2}]
Answer :
Considering the unit conversion formula,
n1U1 = n1U2
n1[M1^{a}L1^{b}T1^{c}] = n2[M2^{a}L2^{b}T2^{c}]
Given here, 1 Cal = 4.2 J = 4.2 kg m^{2} s^{–2}.
n1 = 4.2, M1 = 1kg, L1 = 1m, T1 = 1 sec
and
n2 = ?, M2 = α kg, L2 = βm, T2 = γ sec
The dimensional formula of energy is = [M^{1}L^{2}T^{-2}]
⇒ a = 1, b =1 and c = -2 Putting these values in above equation,
n2= n1[M1/M2]^{a}[L1/L2]^{b}[T1/T2]^{c}
= n1[M1/M2]^{1}[L1/L2]^{2}[T1/T2]^{-2}
= 4.2[1Kg/α kg]^{1}[1m/βm]^{2}[1sec/γ sec]^{-2} = 4.2 α^{–1 }β^{–2} γ^{2}
Answer :
Dimensions of Kinetic energy K = [M^{1}L^{2}T^{-2}]
Dimensions of Moment of Inertia (I) = [ (2/5)Mr^{2}] = [ML^{2}T^{0}]
Dimensions of angular speed ω = [θ/t] = [M^{0}L^{0}T^{-1}]
Applying principle of homogeneity in dimensions in the equation K = kI^{a}ω^{b}
[M^{1}L^{2}T^{-2}] = k ( [ML^{2}T^{0}])a([M^{0}L^{0}T^{-1}])b
[M^{1}L^{2}T^{-2}] = k [M^{a}L^{2a}T^{-b}]
⇒ a = 1 and b = 2
⇒ K = kIω^{2}
Question 63. Convert 1 Newton Into Dyne Using Method Of Dimensions?
Answer :
Dimensions of Force = [M^{1}L^{1}T^{-2}]
Considering dimensional unit conversion formula i.e. n1[M1^{a}L1^{b}T1^{c}] = n2[M2^{a}L2^{b}T2^{c}]
⇒ 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.(10^{3}/1g)(10^{2}cm) = 10^{5}dyne
Answer :
Given, F ∝ m^{a}.v^{b}.r^{c}
∴ F = km^{a}.v^{b}.r^{c } (where k is constant)
Putting dimensions of each quantity in the equation,
[M^{1}L^{1}T^{-2}] = [M^{1}L^{0}T^{0}]^{a}.[M^{0}L^{1}T^{-1}]^{b}. [M^{0}L^{1}T^{0}]^{c} = [M^{a}L^{b+cT}+cT^{-b}]
⇒ a =1, b +c = 1, -b = -2
⇒ a= 1, b = 2, c = -1
∴ F = km^{1}.v^{2}.r^{-1}= kmv^{2}/r
Answer :
Given,
c = 3 x 10^{10} cm/sec
G = 6.67 x 10^{8 }dyn cm^{2}/gm^{2}
h = 6.6 x 10^{-27} erg sec
Putting respective dimensions,
Dimension formula for c = [M^{0}L^{1}T^{-1}] = 3 x 10^{10 }cm/sec …. (I)
Dimensions of G = [M^{-1}L^{3}T^{-2}] = 6.67 x 10^{8}dyn cm^{2}/gm^{2} …(II)
Dimensions of h = [M^{1}L^{2}T^{-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,
⇒ [M^{0}L^{1}T^{-1}].[M^{1}L^{2}T^{-1}].[M^{1}L^{-3}T^{2}]
= ( 3 x 10^{10 }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.
[M^{-1}L^{3}T^{-2}].[M^{1}L^{2}T^{-1}].[M0L-3T3]
= (6.67 x 10^{8 }dyn cm^{2}/gm^{2} ).( 6.6 x 10^{-27}erg sec)/(3 x 10^{10} cm/sec)^{3}
⇒ [L^{2}] = 1.6304 x 10^{-65}cm^{2}
⇒ [L] = 0.4038 x 10^{-32} cm
or 1cm = [L]/ 0.4038 x 10^{-32} = 2.47 x 10^{-32}unit of length
In eqn-I, [M^{0}L^{1}T^{-1}] = 3 x 10^{10}cm/sec
⇒ [T] = [L] ÷ 3 x 10^{10}cm/s
⇒ [T] = 0.4038 x 10^{-32} cm ÷ 3 x 10^{10}cm/s = 0.1345 x 10^{-42} s
or 1s = [T]/0.1345 x 10^{-42}s = 7.42 x 10^{42}unit of time
Answer :
Given, v = Xt^{2 }+ Yt +Z
Dimensions of velocity v = [M^{0}L^{1}T^{-1}]
Applying applying principle of homogeneity in dimensions, terms must have same dimension.
[v] = [Xt^{2}] + [Yt] + [Z]
∴ [v] = [Xt^{2}]
⇒ [X] = [v] /[t^{2}] = [M^{0}L^{1}T^{-1}] / [M^{0}L^{0}T^{2}] = [M^{0}L^{1}T^{-3}] ….(i)
Similarly, [v] = [Yt]
⇒ [Y] = [v] / [t] = [M^{0}L^{1}T^{-1}]/ [M^{0}L^{0}T^{-1}] = [M^{0}L^{1}T^{-2}] …(ii)
Similarly, [v]= [Z]
[Z] = [M^{0}L^{1}T^{-1}] …(iii)
⇒ Unit of X = m-s^{-3}
⇒ Unit of Y = m-s^{-2}
⇒ Unit of Z = m-s^{-1}
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] = [ML^{2}T^{-2}] ——— (II)
Putting values of I and II,
[C] = ([AT])^{2}/ [ML^{2}T^{-2}] = [M^{-1}L^{-2}T^{2+2}A^{2}] = [M^{-1}L^{-2}T^{4}A^{2}]
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.
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-4A^{2}
Impulse = Force ✕ time = FVA^{-1}
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