Applying the concept
1. Which one of the following is NOT a unit in SI system?
A ) Farad B ) Pascal C) Poise D) Tesla
 Analysis: farad is SI unit of capacity of a conductor. pascal is SI unit of pressure. tesla is SI unit of magnetic induction. poise means dyne s cm^{2}. Thus poise is a cgs unit.
 Ans: Thus poise is not a unit in SI system.
2. Length cannot be measured in terms of ………
A) Fermi B ) Micron C) Bar D) Astronomical unit
 Analysis: Fermi, micron, and astronomical unit are units of measurement of length while the bar is a unit of pressure.
 Ans: Length cannot be measured in terms of bar
3. If x = at + bt^{2}, where x is the distance traveled by the body in kilometre while t is the time in second then the unit of b is …
A) km s^{2} B) km s C) km/s^{2} D) km/s
 Analysis: By the principle of homogeneity the net unit on either side of the physical equation must be the same. The unit on the lefthand side is kilometre, hence the unit of the righthand side must be kilometre. Now two physical quantities can be added or subtracted if and only if their units are the same. Thus the unit of (at) and unit of (bt^{2}) should be kilometer.
unit of b x (unit of t)^{2} = kilometer
unit of b = km/s^{2}
 Ans: The unit of b is km/s^{2}.
4. Which one of the following units is correctly expressed?
A ) Metre B ) Newton C) newton D) Joule/s
 Analysis: All units should start with small case letters. If the name of the unit is the name of the scientist, then if written full should start with small case letter and if written in symbol form should be in capital letter. Metre is not correct. The correct representation is metre or m. Newton is not correct. The correct representation is newton or N, Joule/s is not correct. The correct representation is joule/s or J/s. Hence in the given set newton is correctly expressed unit.
 Ans: newton is expressed correctly as per guidelines.
5. The unit 1 N/m is equivalent to ……
A) 1 erg/cm B) 1erg / cm^{2} C) 1 J/m D) 1 J/m^{2}
 Analysis: Dimensions of quantity having unit N/m are [L^{1}M^{1}T^{2}]/[L^{1}] =[L^{0}M^{1}T^{2}]. Dimensions of quantity having unit erg/cm and J/m are [L^{2}M^{1}T^{2}]/[L^{1}] =[L^{1}M^{1}T^{2}]. As these dimensions are not same as that of given quantity. Hence option A and C are ruled out.
 Dimensions of quantity having unit erg/cm^{2} and J/m^{2} are [L^{2}M^{1}T^{2}]/[L^{2}] =[L^{0}M^{1}T^{2}]. Hence one option from B and C is correct.
 Ans: The unit 1 N/m is equivalent to 1 J/m^{2}
6. How many metre are there in a lightyear?
A) 8.5 x 10^{14 }m B) 9.8 x 10^{15} m C) 9.46x 10^{15} m D) 9.8 x 10^{14} m
 Analysis: 1 lightyear = distance travelled by light in 1 year
1 lightyear = 3 x 10^{8} x 365 x 24 x 60 x 60 m = 9.46 x 10^{15} m
 Ans: There are 9.46 x 10^{15} m in a lightyear
7. Joule / cm^{3} can be a unit of ……….
A ) Impulse B) Force C) Pressure D) Momentum
 Analysis: Dimensions of quantity having unit joule/cm^{3} is [L^{2}M^{1}T^{2}]/[L^{3}] = [L^{1}M^{1}T^{2}]. Dimensions of impulse are [L^{1}M^{1}T^{1}]. Dimensions of force are [L^{1}M^{1}T^{2}]. Dimensions of pressure are [L^{1}M^{1}T^{2}]. Dimensions of momentum are [L^{1}M^{1}T^{1}]. As dimensions of pressure are same as that of given quantity, Joule / cm^{3} can be a unit of pressure.
 Ans: Joule / cm^{3} can be a unit of pressure.
8. In a equation { P + a / V^{2} } ( V – b) = constant , where P is the pressure, V is the volume and a, b are constants, then unit of ‘a’ is …….
A ) dyne cm^{5} B) dyne cm^{4} C) dyne cm^{3} D) dyne cm^{2}
 Analysis: Two physical quantities can be added or subtracted if and only if their units are the same.
Thus the units of P and a / V^{2} are same
Unit of P = Unit of a /(unit of V)^{2}
dyne cm^{2} = unit of a / (cm^{3})^{ 2}
dyne cm^{2} = unit of a / cm^{6}
unit of a = dyne cm^{2} x cm^{6 }= dyne cm^{4}
 Ans: The unit of a is dyne cm^{4}
9. An error is committed by a student in removing the parallax is ………
A) an instrumental error B ) a personal error C) systematic error D) a random error
 Analysis: In this case, the error is due to the personal problem of the student. It can be minimized by using a proper reading technique by the student. Hence it is a personal error.
 Ans: Error is committed by a student in removing the parallax is a personal error
10. Zero error is included in the category of ………………
A) an instrumental error B) a personal error C ) systematic error D ) accidental error
 Analysis: Zeroerror is a type of error in which an instrument gives a reading when the true reading at that time is zero. For example needle of ammeter failing to return to zero when no current flows through it. Thus the zero error is caused by the defect in the instrument. Hence it is an instrumental error.
 Ans: Zeroerror is included in the category of an instrumental error
11. A force F is given by F = at + bt^{2}, where t is the time. The dimensional formula of ‘a’ and ‘b’ are ……….
A) [M^{1}L^{1}T^{3}] and [M^{1}L^{1}T^{4}] B ) [M^{2}L^{1}T^{2} ] [ M^{1}L^{1}T^{2}] C) [ M^{1}L^{1}T^{2}] [M^{1}L^{2}T^{3}] D ) [M^{1}L^{1}T^{4}] [M^{1}L^{1}T^{3}]
 Analysis: By the principle of homogeneity the dimensions on either side of the physical equation must be the same. Now two physical quantities can be added or subtracted if and only if their dimensions are the same.
Dimensions of F = Dimensions of at
[L^{1}M^{1}T^{2}] = [a] [t]
[a] =[L^{1}M^{1}T^{2}] / [T]
[a]=[L^{1}M^{1}T^{3}]
Dimensions of F = Dimensions of bt^{2}
[L^{1}M^{1}T^{2}] = [b] [T]^{ 2}
[b]= [L^{1}M^{1}T^{2}] / [T^{2}]
[b]= [L^{1}M^{1}T^{4}]
 Ans: The dimensional formula of ‘a’ and ‘b’ are [M^{1}L^{1}T^{3} ] and [M^{1}L^{1}T^{4}].
12. Solar constant (S) is defined as the amount of solar energy received per cm^{2} per minute. What is the dimensional formula for solar constant?
A ) [M^{0}L^{0} T^{2}] B ) [M^{1}L^{0}T^{3} ] C ) [M^{1}L^{1}T^{2}] D) [M^{2}L^{0}T^{2}]
 Analysis: Given S = E/At
 Ans: The dimensional formula for the solar constant is [M^{1}L^{0}T^{3}]
13. According to Laplace’s formula, the velocity ( V) of sound in a gas is given by v = (γP / d)^{1/2}, where P is the pressure, d is the density of a gas. What is the dimensional formula for γ?
A ) [ M^{1} L^{1} T^{1} ] B ) [ M^{1} L^{0} T^{1}] C ) [ M^{1} L^{0} T^{1}] D) [ M^{0} L^{0} T^{0} ]
 Analysis:
 Ans: The dimensional formula for γ is [ M^{0} L^{0} T^{0} ]
14. Bernoulli’s equation is given by P + 1/2 d v^{2} + hdg = K, where P,d,V,h, and g have their usual meanings. The dimensional formula for K is same as that of ……..
A ) thrust B ) pressure C) pressure gradient D) velocity gradient.
 Analysis: Two physical quantities can be added or subtracted if and only if their units are the same. One of the quantity on LHS is pressure. Thus all other quantities on RHS should have same dimensions as that of pressure. Hence LHS should have dimensions of pressure. Hence K should have dimensions of pressure.
 Ans: The dimensional formula for K is same as that of pressure.
15. The unit of solid angle is steradian, what is the dimensional formula of steradian?
A ) [M^{1}L^{1}T^{1}] B) [M^{0}L^{0}T^{0}] C) [M^{1}L^{2}T^{1}] D ) [M^{1}L^{2}T^{0}]
 Analysis: Steradian is dimensionless quantity. Hence its dimensions are [ M^{0} L^{0} T^{0} ]
 Ans: The dimensional formula of a steradian is [ M^{0} L^{0} T^{0} ]
16. In an equation { P + a / V^{2} } (V – b) = RT, where P is the pressure, V is the volume, T is the temperature and a, b, R are constants. What is the dimensional formula of a /b?
A ) [ M^{2} L^{2} T^{2} ] B) [ M^{1} L^{1} T^{2} ] C) [ M^{1} L^{2} T^{2} ] D ) [ M^{2} L^{3} T^{1} ]
 Analysis: Two physical quantities can be added or subtracted if and only if their dimensions are the same.
Thus the dimensions of P and a / V^{2} are same
Dimensions of b = Dimensions of V
[b] = [L^{3}M^{0}T^{0}]
Now [a]/[b] = [L^{5}M^{1}T^{2}]/ [L^{3}M^{0}T^{0}] = [L^{2}M^{1}T^{2}]
 Ans: The dimensional formula of a /b is [ M^{1} L^{2} T^{2} ]
17. Which one of the following group have quantities that DO NOT have same dimensions ?
A ) velocity, speed B) pressure, work/ volume C ) force, impulse D ) work, energy
 Analysis: By definition impulse = force x time. Thus force and impulse do not have same dimensions.
 Ans: Force and impulse do not have same dimensions.
18. Which of the following operation of the functions of A and B can be performed if A and B possess different dimensions?
A) A+B B) A – B C) A/B D ) [(A+B)/(AB)]
 Analysis: Two physical quantities can be added or subtracted if and only if their dimensions are the same. It is given that A and B possess different dimensions. Thus options A, B, and D are incorrect. If two quantities have different dimensions, then they can be multiplied or be divided. Thus option C is correct.
 Ans: A/B operation of the functions of A and B can be performed
19. A force is given in terms of distance x and time t by F = A sin Ct + B cos Dx. Then what are the dimensions of A/B and C/D?
A ) [M^{0}L^{0}T^{0}] [M^{0}L^{0} T ^{1}] B) [M^{1}L^{1}T^{2}] [M^{0} L^{1} T^{0}] C) [M^{0}L^{0}T^{0}] [M^{0}L^{1} T ^{1}] D) [M^{0}L^{1}T^{1}] [M^{0} L^{0} T^{0}]
 Analysis: By the principle of homogeneity the dimensions on either side of a physical equation must be the same. The trigonometric ratios sin and cos are pure ratios of length, hence are dimensionless quantity.
Thus [A] = [B] =[F] = [ M^{1} L^{1} T^{2} ] Thus dimensions of A/B are [ M^{0} L^{0} T^{0} ]
Now the angle is dimensionless quantity
[C][t] = [ M^{0} L^{0} T^{0} ]
[C] = [ M^{0} L^{0} T^{0} ]/ [T] = [ M^{0} L^{0} T^{1} ]
[D][x] = [ M^{0} L^{0} T^{0} ]
[D] = [ M^{0} L^{0} T^{0} ]/ [L] = [ M^{0} L^{1} T^{0} ]
[C]/[D] = [ M^{0} L^{0} T^{1} ]/ [ M^{0} L^{1} T^{0} ] = [ M^{0} L^{1} T^{1} ]
 Ans: The dimensions of A/B and C/D are [M^{0}L^{0}T^{0}] [M^{0}L^{1} T^{1}]
20. The dimensions of coefficient of viscosity are ……….
A) [M^{1}L^{2}T^{2}] B) [M^{1}L^{1}T^{1}] C) [M^{1}L^{1}T^{1}] D ) [M^{1}L^{1}T^{2}]
 Ans: The dimensions of coefficient of viscosity are [M^{1}L^{1}T^{1}]
21. The SI unit of universal gravitational constant G is ……………..
A) Nm kg^{2} B) Nm^{2} kg^{2} C) Nm^{2} kg^{1} D) Nm kg^{1}
 Ans: The SI unit of universal gravitational constant G is Nm^{2} kg^{2}
22. In a equation { P + a / V^{2}} (V —b) = nRT, where P is the pressure, V is the volume , T is the temperature and a, b, R are constants. The dimensions of nRT are same as that of ……….
A ) energy B ) force C) pressure D ) specific heat
 Analysis: Two physical quantities can be added or subtracted if and only if their dimensions are the same. Thus dimensions of { P + a / V^{2} } are that of pressure while dimensions of (V — b) are that of volume
[P][V] =[nRT]
[nRT] = [L^{1} M^{1}T^{2}] [L^{3} M^{0}T^{0}] = [L^{2} M^{1}T^{2}]
These are the dimensions of energy.
 Ans: The dimensions of nRT are same as that of energy.
23. In a equation { P + a / V^{2} } (V – b) = nRT, where P is the pressure, V is the volume , T is temperature and a, b, R are constants. The dimensions of ‘a’ are same as that of ……………..
A) PV B ) PV^{2} C) P^{2}V D) P/V
 Analysis:
 Two physical quantities can be added or subtracted if and only if their dimensions are the same.
 Ans: The dimensions of ‘a’ are same as that of PV^{2}
24. In a equation {P + a / V2} (V – B) = nRT, where P is the pressure, V is the volume , T is the temperature and a , b ,R are constants . The dimensions of ‘b’ are same as that of …………..
A) P B) V C) PV D) nRT
 Analysis: Two physical quantities can be added or subtracted if and only if their dimensions are the same. Thus dimensions of b and those of V should be same.
 Ans: The dimensions of ‘b’ are same as that of V.
Definitions:
Unit of a physical quantity:
 A unit of a physical quantity is a selected magnitude of a physical variable in terms of which other magnitudes of the same variable can be expressed. OR the reference standard used for measurement of a physical quantity is called the unit of physical quantity.
 Example: metre is a unit of length
Fundamental quantity:
 Fundamental quantity is that quantity which does not depend on other quantities for its measurement.
 Examples: Mass, Length and Time are fundamental quantities.
Fundamental unit:
 The unit of a fundamental quantity called as the fundamental unit.
 Examples: mass, length, time etc. are fundamental quantities. while, their units metre, kilogram, second etc. are fundamental units.
Derived quantity:
 Derived quantity is that quantity which depends on two or more other quantities for its measurement.
 Example: Velocity = displacement/time. Velocity depends on two other quantities namely displacement and time. Hence velocity is a derived quantity. Other examples are density, acceleration, force, momentum, pressure etc.
Derived unit:
 The units of derived quantities are called as derived units
 Examples: density, acceleration, velocity, force, momentum, pressure etc. are derived quantities. while, their units kg m^{3}, m s^{2}, m s^{1}, newton, kg m s^{1}, pascal etc. are derived units.
Dimensional equation:
 An expression, which gives the relation between the derived units and fundamental units in terms of dimensions is called a dimensional equation.
 Example: The dimensional equation of speed is [v] = [L^{1}M^{0}T^{1}]. and 1, 0, 1 are called dimensions.
Dimensions of a physical quantity:
 The power to which fundamental units are raised in order to obtain the unit of physical quantity is called the dimensions of that physical quantity.
 Dimensions of speed are [L^{1}M^{0}T^{1}].
Most probable value:
 When the sufficiently large number of readings are taken, then the mean of these readings is called as most probable value.
Absolute error:
 The magnitude of the difference between the most probable value (mean) and the individual measurement is called the absolute error of the measurement.
Relative error:
 The ratio of the absolute error in the measurement of a quantity with the most probable value is called as relative error
Percentage error:
 If relative error is multiplied by 100, the value obtained is called as percentage relative error.
Significant figures:
 The digits, whose values are accurately known in a particular measurement are called significant figures.
 e.g. in the number 3.400, the number of significant figures is 4 while in 0.0034, the number of significant figures are 2.
Order of magnitude:
 The order of magnitude of a physical quantity is defined as the value of its magnitude rounded off to the nearest integral power of 10.
 For e.g. if a man has a weight of 96 kg, then his weight can be rounded off to the nearest integral power of 10 is 100 kg or 10² kg. Hence the order of magnitude of man is 10² kg.
Solar day:
 A solar day is an interval from one noon to the next noon.
Mean solar day:
 Average of the length of a solar day over a year is considered as a mean solar day.
Knowledge Bits:
 The cesium atomic clocks are very accurate.
 A mass spectrograph is used for measurement of a mass of atomic/subatomic particles.
 A physical quantity has dimensions. Is it necessary it must have a unit?
 If a physical quantity has dimensions then it must have unit because we derive the dimensions by directly or indirectly using the units.
 Tere are some physical quantities having units but no dimensions.examples are plane angle, angular displacement, solid angle.
 There are physical quantities which neither have dimensions nor have units. Examples are pure ratios of similar quantities like strain, specific gravity.
Responsibility of National Physical Laboratory (NPL):
 It is the responsibility of the NPL to calibrate the measurement standards in these laboratories at different levels.
 The weights and balances used in local markets and other areas are expected to be certified by the Department of Weights and Measures of the local government.
 To strengthen and advance physicsbased research and development for the overall development of science and technology in the country.
 To establish, maintain and improve continuously by research, for the benefit of the nation,
 To identify and conduct after due consideration, research in areas of physics which are most appropriate to the needs of the nation and for the advancement of the field
 To assist industries, national and other agencies in their developmental tasks by precision measurements, calibration, development of devices, processes, and other allied problems related to physics.
 To keep itself informed of and study critically the status of physics.
The error caused due to faulty construction of instruments: Instrumental error
The error associated with human error: Personal error
The error due to defective settings of an instrument: Systematic error
Random error: An error in measurement caused by factors which vary from one measurement to another is called random error. Example: error caused in the measurement of the length of a rod due to change in temperature.
Two examples of dimensionless physical quantities: plane angle, angular displacement, solid angle, strain, specific gravity, Poisson’s ratio.
The system of units is universally accepted: SI system of units
Can two different physical quantities have same dimensions? If yes state the example.
Q 13. Priya , Riya and Siya were asked by their teacher to measure the mass of
wooden block. Priya wrote : 200 gm , Riya wrote : 200 g and Siya wrote :
200 gs Which one of these answer is correct ?
Q 14. 1eV = 1.6 X 1019 J. Binding energy of H atom is 13.6 eV, express this energy
in joule.
Q 15. If the volume of a block V = l . b. h. then state the formula for relative error
in V.
Requirements of a good unit:
 It should be well defined without any doubt or ambiguity.
 It should be of suitable size. i.e. neither too long nor too small in comparison with quantity to be measured.
 It should be easily available.
 It should be nondestructible.
 It should not change with the time.
 It should not change with the place.
 It should be easily reproducible.
Fundamental quantities and their two examples with their S. I. Unit.
 Fundamental quantity is that quantity which does not depend on other quantities for its measurement.
 Examples: mass, length, time etc. are fundamental quantities. while, their units metre, kilogram, second etc. are fundamental units.
Derived quantities and their two examples with their S. I. Unit.
 Derived quantity is that quantity which depends on two or more other quantities for its measurement.
 Examples: density, acceleration, velocity, force, momentum, pressure etc. are derived quantities. while, their units kg m^{3}, m s^{2}, m s^{1}, newton, kg m s^{1}, pascal etc. are derived units.
Uses of dimensional analysis:
 To check the correctness of physical equation:
 To Find Dimensions of New Physical Quantity:
 To find the form of a physical equation, we use a physical equation which contains the quantity, whose physical dimensions are to be found.
 To derive the form of a physical equation:
 To derive the relation between different units of different systems of a physical quantity:
Dimensions of following physical quantities:
Velocity (v):
Dimensions of velocity or speed are [L^{1}M^{0}T^{1}]
Acceleration (a or f):
Dimensions of acceleration are [L^{1}M^{0}T^{2}]
Force (F):
Force = Mass × Acceleration
∴ F = m× a
∴ [F] = [m] × [a]
∴ [F] = [L^{0}M^{1}T^{0}][L^{1}M^{0}T^{2}]
∴ [F] = [L^{1}M^{1}T^{2}]
Dimensions of force are [L^{1}M^{1}T^{2}]
Momentum (p):
Momentum = Mass × Velocity
∴ p = m× v
∴ [p] = [m] × [v]
∴ [p] = [L^{0}M^{1}T^{0}][L^{1}M^{0}T^{1}]
∴ [p] = [L^{1}M^{1}T^{1}]
Dimensions of momentum are [L^{1}M^{1}T^{1}]
Impulse (J):
Impulse of Force = Force × Time
∴ J =F× t
∴ [J] = [F] × [t]
∴ [J] = [L^{1}M^{1}T^{2}][L^{0}M^{0}T^{1}]
∴ [J] = [L^{1}M^{1}T^{1}]
Dimensions of impulse of force are [L^{1}M^{1}T^{1}]
Work (W):
Work Done = Force × Displacement
∴ W =F× s
∴ [W] = [F] × [s]
∴ [W] = [L^{1}M^{1}T^{2}][L^{1}M^{0}T^{0}]
∴ [W] = [L^{2}M^{1}T^{2}]
Dimensions of work are [L^{2}M^{1}T^{2}]
Energy (E or U):
Potential Energy = Mass × Acceleration due to gravity × Height
∴ E =m× g × h
∴ [E] = [m] × [g] × [h]
∴ [E] = [L^{0}M^{1}T^{0}][L^{1}M^{0}T^{2}][L^{1}M^{0}T^{0}]
∴ [E] = [L^{2}M^{1}T^{2}]
Dimensions of energy are [L^{2}M^{1}T^{2}]

Power (P):
Dimensions of Power are [L^{2}M^{1}T^{3}]
Pressure (P):
Dimensions of Pressure are [L^{1}M^{1}T^{2}] S.I. Unit of pressure is
Velocity gradient
Thermal conductivity
Magnetic field
Universal constant of gravitation:
 If m_{1} and m_{2} are two masses separated by a distance r from each other then the force of gravitation acting between them is given by Newton’s law of gravitation
Hence dimensions of universal gravitation constant are [L^{3}M^{1}T^{2}]
Coefficient of Viscosity:
 Le F be the viscous force acting between two layers of liquid area A having velocity difference of dv between them. Let dx be the separation between the two layers and η is coefficient of viscosity, then by Newton’s law of viscosity
Hence dimensions of the coefficient of viscosity are [L^{1}M^{1}T^{1}]
Method to Minimize the effect of errors:
 Errors can be minimized by taking a number of readings and then finding the average of the readings taken.
Rules for determining significant figures:
 One and only one uncertain digit is to be retained in a measurement. for e.g. When we record a reading as 2.65 it means that we are sure of first two digits but we are not sure about the last digit Hence there is only one uncertain digit
 When a number is to be rounded off to a specific number of significant figures then, If number dropped is less than five, the last digit retained is left unchanged. for e.g. If the number is 12.42 is to be rounded to three significant figures then it is rounded off as 12.4. If number dropped is equal to or greater than five, the last digit retained is increased by 1. for e.g. If the number is 17.49 is to be rounded to three significant figures then it is rounded off as 17.5
 All nonzero digits are significant. All zeros occurring between two nonzero digits are significant. The zeros on the left side of the number are not significant. for e.g. in the number 0.00034, the number of significant figures is 2.
 The zeros on the right side of the number are significant because they indicate the accuracy of the measurement. for e.g. in the number 3.400, the number of significant figures is 4.
 All zeros to the right of the last rightmost nonzero digits are not significant. For e.g. in 2020 there are 3 significant figures.
 All zeros to the right of the last nonzero digit are significant if they come from the measurement. For e.g. If the distance between two points is measured 2020 cm (measured to nearest centimeter) then in this number there are four significant figures.
 If the number contains more digits than the significant figures, the number should be expressed as a power of 10. for e.g. earth mass is known to be correct up to 3 significant figures, hence it is expressed as 5.98 × 10^{24} kg.
Determination of significant figures of the following measurements:
Sr. No.  Number  Significant Figures 

1  0.07787  4 
2  11.2  3 
3  1.6 X 10 ^{19}  2 
4  3 X 10^{8}  1 
Order of magnitude of the following quantities:
Acceleration due to gravity at the pole is ‘g’ = 9.83m /s^{2}.
Acceleration due to gravity = 9.83 m s^{2} = 9.83 × 10^{0} m s^{2}
Hence the order of magnitude of acceleration due to gravity is 10^{1}.
Universal constant of gravitation ‘G’ = 6.67 X 10^{11} Nm^{2}/ kg^{2}.
Universal constant of gravitation ‘G’ = 6.67 X 10^{11} Nm^{2}/ kg^{2}.
Hence the order of universal constant of gravitation is 10^{10}.
If the force (F), acceleration (A), and time (T) are taken as fundamental Units, then find the dimensions of energy.
We have Force = Mass × Acceleration
∴ Mass = Force / Acceleration
∴ [Mass] = [F^{1}] / [A^{1}] = [F^{1}A^{1}]
We have Acceleration = Velociy/ time
∴ Acceleration = (distance/time)/ time
∴ Acceleration = distance/time^{2}
∴ distance = acceleration x time^{2}
∴ [Length] = [Acceleration][time]^{2} = [AT^{2}]
Now, energy = mass x gravitational acceleration x height
∴ [Energy] = [mass] x [acceleration] x [length]
∴ [Energy] = [F^{1}A^{1}] x [A^{1}] x [AT^{2}]
∴ [Energy] = [F^{1} A^{1} T^{2}]
Dimensions of energy are [F^{1} A^{1} T^{2}]
The velocity (v) of water waves may depend on their wavelength (λ) density of water (d) and acceleration due to gravity ( g ). Find the relation between these quantities by using dimensions.
let v ∝ λ^{x}, v ∝ d^{y}, v ∝ g^{z},
Combining above relations we have v ∝ λ^{x }d^{y }g^{z},
v = k λ^{x }d^{y }g^{z}^{ }………….. (1)
By principle of homogeneity of dimensions we have
[v] = [λ]^{x }[d]^{y }[g]^{z }
∴ [L^{1}M^{0}T^{1}] =^{ }[L^{1}M^{0}T^{0}] ^{x} [L^{3}M^{1}T^{0}]^{y }[L^{1}M^{0}T^{2}]^{z }
∴ [L^{1}M^{0}T^{1}] =[L^{x}M^{0}T^{0}]^{ }[L^{3y}M^{y}T^{0}]^{ }[L^{z}M^{0}T^{2z}]
∴ [L^{1}M^{0}T^{1}] = [L^{x3y + z }M^{y}T^{2z}]
Considering equality of two sides we have
x – 3y + z = 1 , y = 0 , 2z = 1 i.e. z = 1/2
∴ x – 3y + z = 1
∴ x – 3(0) + 1/2 = 1
∴ x = 1 – 1/2 = 1/2
Substituting x =1/2, y = 0 and z = 1/2 in equation (1) we get
v = k λ^{1/2 }d^{0 }g^{1/2}^{ }
∴ v = k (λ g)^{1/2}^{ }
∴ v = k √λ g^{ }
This is the form of equation for a velocity of a wave.
Conversion factor between S. I. Unit and C.G.S. unit of following quantities
Force:
newton and dyne are units of force in S.I. and c.g.s. system respectively
Dimensions of force are [L^{1}M^{1}T^{2}]
_{}
Energy:
joule and erg are units of energy in S.I. and c.g.s. system respectively
Dimensions of force are [L^{2}M^{1}T^{2}]
(C) Density (D) Pressure.
Q 4. Young’s modulus of steel is 19 x 1010 N / m2. Express it in dyne / cm2 .
Q 5. State which of the following is dimensionally correct ?
A) Pressure = Energy per unit volume.
B) Pressure = Momentum x Volume x Mass
Q 6. Verify the correctness of the following equations dimensionally. (each
equation carry 3 marks) (Symbols have their usual meaning)
A ) s = ut + at2
B ) Ft = mv
 C) F = m v2/ r
 D) P = h p g
 E) F = 6 η r v
 F) σ = q/A
 G) ρ = q/v (ρ is volume charge density)
H) E = F / qo ( E is electric field )
I) P = q X 2 l ( P is electric dipole moment )
Q 7. In a equation { P + a / V2} ( V — b) = RT, where P is the pressure, V is the
volume, T is the temperature and a, b, R are constants. Find the
dimensions of (each sub question 3 marks)
A ) a and b B ) a/b C) a. b
Q 8. What do you mean by an error ? Explain (a) Instrumental error ( b ) Systematic
(persistant) error .
Q 9. What is the cause of an error? Explain a) personal error b) random error
S.A.I (2 Marks)
 Determine the number of significant figures in the following measurements :
(a) 0.002901 (b) 980 (c) 6.63 X 10 34 (D) 98.00
2. State the order of magnitude of the following :
(a) 8.85 X 10 12 Wb / Am (b) 1.6 X 1019 C
(c) 6400 km (d) 0.00927
3. What is the order of magnitude of one light year expressed in metre ?
4. Two carbon resistances are given by R1= ( 4 + 0.4 ) ohm and R2 = ( 12 + 0.6 ) ohm.
Calculate the percentage error , if they are connected in series.
5. If the velocity (v ) , time ( t ) and force (f) were chosen as fundamental quantities then
obtain the dimensional formula of mass.
S.A.II (3 Marks)
1. In the following equation , x, t, and f represent displacement , time and force
respectively .
f = a + bt + Asin ( w + f ) + 1 / ( c + xd ) . What is the dimensional formula for A.c ?
2. A metal wire has a mass ( 0.5 + 0.005 ) g , radius ( 4 + 0.004 ) mm and length ( 5 +
0.05) cm. Calculate the percentage error in the measurement of its density.
3. A student performing an experiment to find the period of second’s pendulum
obtained the following results :
1.96 s , 2.00 s , 1.94 s , 2.04 s and 2.06 s . Find an absolute error , relative error and
percentage error in the measurement.