Chapter – 2
Units and Measurements
Physical quantity: A quantity which can be measure directly or indirectly is called a physical quantity. E.g.: Mass of a body, length of an object, velocity, acceleration, time of an events etc.
Units: Unit is a standard quantity of the same kind with which a physical quantity is compared for measuring it i.e. physical quantity = number x units.
E.g.: Mass of a body 4/5 kg. And Length of an body
Mass: Mass of a body is defined as the quantity of matter constant in the body. Mass of a body is defined in two different types:
 Inertial mass.
 Gravitational mass.
Inertial Mass: Inertial mass of an body is the quantity measure of Inertia of the body.
Gravitational mass: Gravitational mass of a body is its property on which the gravitational force of attraction of earth on the body depends.
Notes: (1) Both gravitational mass and inertial mass are equal. (ii) Since both gravitational mass and inertial mass are equal, so a single term “mass” is used to describe both the form of mass.
Unit of mass is Kilogram (Kg): 1 Kg was defined as the mass ofcubic metre of water at 4 0C.
Length: The extend of separation or the distance between two points in space is called length. Unit of length is metre.
Time: According to Einstein, time is simply what a clock reads. In facts time is the measure of the duration between the occurrences of two events or time is measure of the duration for which an events lash.
The S.I. unit of time is second.
Fundamental Units:

S.I. Units

(i) Length
(ii) Mass
(iii) Time
(iv) Electric Current
(v) Temperature
(vi) Luminous Intensity
(vii) Amount of substances

Metre (m)
Kilogram (Kg)
Second (S)
Ampere (A)
Kelvin (K)
Candela (Cd)
Mole (mol)

Supplementary Units: (i) Plane Angle, (ii) Solid angleboth are unit less quantity.
System of Errors:
 Systematic error: The systematic error are those errors that tends to be in one direction, either positive or negative.
 Instrumental errors: Instrumental errors that arise from the errors due to imperfect design or calibration of the measuring instruments.
 Personal errors: Personal errors that arise due to lack of proper setting of the apparatus of Carolinas in talking observation without observing proper precautions.
 Least count error: The smallest value that can be measured by the measuring instrument is called least count error. The least count error is the error associated with theof instruments.
Significant figures:
 All the non zero digits are significant.
 All the zeroes between two nonzero digits are significant, no matter where the decimal point, if at all.
Conversion of angles in degree, minutes and second into radian:
Supplementary Units:
 Radian (Rad): It is the unit of plane angle. One radian is an angle subtended at the centre of a circle by an arc of length the radius of the circle.
 Ste radian (Sr): It is the units of solid angle. One Ste radian is the solid angle subtended at the centre if a sphere by its surface whose area = the square of the radius of the sphere.
For large measurement:
 Astronomical unit: the average distance between the earth and the sum is called astronomical unit. i.e.
 Light year (ly): The distance travelled by light in vacuum in 1 year is called light year. i.e.
 LASSER: Light Amplification by stimulated Emission of Radiation.
 MATER: Microwave Amplification by stimulated Emission of Radiation.
It is used to determine the distance of earth from the moon.
Fundamental Physical quantity: The physical quantities independent of each other are called fundamental physical quantity.
Fundamental units: The units in which fundamental physical quantities are expressed are known as fundamental units.
 The largest unit of mass is Chandra Shaker limit = 1.4 times the mass of the sun.
 The nearest star to our solar system is alpha century.
Q. Define 1 metre.
Ans. 1 Metre = distance travelled by time in vacuum inof a sound.
Q. Define 1 second.
Ans. One second = time taken by 9, 192, 631, 770. Vibration of calcium – 133 atom.
 Size of atom is usually measured in astronomy
 Size of an nucleus of an atom is measured in Term
 Parsec: The distance of a point at which an are of length equal to 1 astronomical unit subtended an angle of 1 second.
1 Parsec =
1. Differentiation:
Q.
E.g.:
Q.
Q.
Q.
Q.
Q.
Q.
Dimension Analysis: Dimension of a physical quantity are the powers to which the fundamental quantities are to be raised to represent that physical quantity.
Dimensional formula: The expression which shows that which of the fundamental quantities and with what powers enter into the derived unit of physical quantity is known as dimensional formula of that physical quantity. For e.g. dimensional formula for area is
Principle of homogeneity: According to this principle the dimension of the fundamental quantities of two sides of a physical relation must be same.
 Length =
 Volume =
 Time =
 Density =
 Velocity =
 Linear momentum =
 Acceleration =
 Force =
 Work = = energy
10. Kinetic energy:
11. P.E.
12. Angle =
Q. Check the dimensional consistency in case of the following equation:
By the principle of longevity the dimension of fundamental unit of each term is same so the relation is dimensionally correct.
Q. Derive dimensionally the expression for the centripetal force acting on a body of mass (m) moving with a velocity (v) in a circle of radius (r).
Ans. Here, force (f) depends upon
 Mass (m) of the particle.
 Velocity (v) of the particle
 Radius (r) of the circle.
i.e.
Now, comparing the powers of fundamental quantities
Or
By principle of homogeneity we gets
Q.
Q. Check the correction of the following relationusing dimensional analysis.
Ans. Velocity,
G, Gravitional constant
Now,
Since the dimension of the fundamental units on both the sides are same, hence the given relation is correct.
Q. Suppose that the aukilation of a simple pendulum depends on:
 Mass of the pendulum (m).
 The length of the shring (l).
 Acceleration due to gravity (g).
 Angular displacement (Q).
Dimensionally show which of the above factors have an influence upon the time period and in what way.
Ans. Here the ackilaration of a simple pendulum depend upon:
 Mass (m) of the pendulum.
 Length (L) of the shring.
 Acceleration (g) due to gravity.
 Angular displacement (Q).
i.e.
By the principle of Homogeneity,
Thus the time period of the pendulum is directly proportional to the square root of the length of the shring and inversely proportional to the square root of acceleration due to gravity and is independent of the mass and its angular displacement.
Q. Check the correctness of the following relation:
Solution:
Solution:
Solution:
Q. The force (F) acting on the body depends upon:
 Mass of the body.
 Acceleration of the body.
Find the expression for the force (f) using the method of dimension.
Solution:
Q. The Kinetic energy possessed by a body depends upon its:
 Mass.
 Speed (v)
Find the expression for kinetic energy of the body.
Solution:
Q. A body of mass (m) moving in a circle of radius r with angular velocity (w) finds the expression for the centripetal force.
Solution:
Q. The orbital velocity (v) of the satellite may depend on its mass (m), distance r from the centre of the earth and acceleration due to gravity g. final expression for velocity (v).
Q. Derive by method of dimension and expression for the escape velocity of a body, assuming that in depends upon radius of the planet R and acceleration due to gravity.
Solution:
Here the force of body depend on