Measurement is a process of determining how large or small a physical quantity is as compared to a basic standard. This reference standard is called the unit of the particular physical quantity.

Units of fundamental and derived quantities are known as the fundamental units and derived units, respectively. A complete set of these units, both fundamental and derived units is known as the system of units.

There are some systems used in units, can be defined as
`(1) text(CGS System)` (Centimetre, Gram, Second) is often used in scientific work. This system measures, length in centimetre (cm), mass in gram (g) and time in second (s).
`(2) text(FPS System)` (Foot, Pound, Second) It is also called the British Unit System. This unit measures, length in foot (foot), mass in gram (pound) and time in second (s).
`(3) text(MKS System)` (Metre, Kilogram, Second) This system measures length in metre (m), mass in kilogram (kg) and time in second (s).
`(4) text(SI Units)` (International System of Units) A variety of system of units (CGS, FPS and MKS) leads to the need of a unique system of units which is accepted world-wide. So, in 1971, a system of units named SI (System International in French) was developed and recommended by general conference on weights and measures. It is an extended version of the MKS system.
SI system has seven fundamental units and two supplementary units, which are as follows-

The two supplementary units of SI system are
`(i) text(Radian for Plane Angle)` Angle subtended by an arc at the centre of the circle having length equal to radius of circle has unit radians. It is denoted by `rad`.
`(ii) text(Steradian for Solid Angle)` It is the solid angle which has the vertex at the centre of the sphere and cut-off an area of the surface of sphere equal to that of square with sides of length equal to radius of sphere. It is expressed in unit steradian and denoted by `sr`.

`text(Units Used for Measuring Small Distances)`
`*` 1 cm = `10^-2 m`
`*` 1 mm = `10^-3m`
`*` 1 micron = `10^-4 cm = 10^-6 m`
`*` 1 nanometer = `10^-7 cm = 10^-9 m`
`*` 1 angstrom (`overset@A`) = `10^-8 cm = 10^-10 m`
`*` 1 fermi = `10^-13 cm = 10^-15 m`

`text(Units Used for Measuring large Distances)`
`*` 1 Light year = `9.46xx10^15 m`
`*` 1 Parsec = `3.08xx10^16 m` = 3.26 light year
`*` 1 Astronomical unit (AU) = `1.469xx10^11m`

The dimension of a physical quantity are the power to which the base quantities are raised to represent that quantity. The expression which shows how and which base quantities represent the dimensions of a physical quantity, is called the `text(dimensional formula)`. e.g. for volume, dimensional formula is `[M^0L^3T^0]`. An equation, where a physical quantity is equated with its dimensional formula is called dimensional equation. e.g. dimensional equation for force is `[F]=[MLT^-2]`

S.No	Fundamental Quantity	Dimension
1	Length	[L]
2	Mass	[M]
3	Time	[T]
4	Electric Current	[A]
5	Thermodynamic temperature	[ K ]
6	Luminous intensity	[cd]
7	Amount of substance	[mol]

S.No	Physical Quantity with Formula	Dimensional Formula	SI Units
1	`text(Velocity) = text(displacement)/text(time)`	`[L]//[T] = [M^0LT^-1] `	m/s
2	`text(Acceleration) = text(velocity)/text(time)`	`[LT^-1]//[T] = [M^0LT^-2] `	`m//s^2`
3	`text(Force ) = text( mass x acceleration)`	`[M][LT^-2] = [ ML^-2] `	` kg-m//s^2->` Newton `-> ` N
4	`text(Work) = Fs cos theta`	`[MLT^-2][L] = [ML^2T^-2] `	`kg-m^2//s^2->` joule `-> ` J
5	`text(Kinetic energy) = 1/2 mv^2 `	`[M] [LT^-1]^2 =[ ML^2T^-2] `	joule
6	`text(Potential energy)`	`[M][LT^-2][L] = [ML^2T^-2] `	joule
7	`text(Torque = Fr sin theta `	`[MLT^-2][L] = [ ML^2T^-2]`	N-m
8	`text( Power) = text(work)/text(time)`	`[ML^2T^-2]//[T]`	`kg-m^2//s^3 ->` J//s `-> ` watt `-> `W
9	`text(Momentum = mass x velocity)`	` [M][LT^-1] = [MLT^-1] `	`kg-m//s ` or N-s
10	`text(Impulse) = F Delta t`	`[MLT^-1][T] = [MLT^-1] `	N-s
11	`text(Angle) = text(arc)/text(radius)`	`[L]/[L] = [M^0L^0T^0]`	radian `->` rad
12	`text(Strain) = Delta L//L` or `(DeltaV)/V `	` [L]/[L]` Dimensionless	no unit
13	`text(Frequency) = 1/text( time period)`	`1/[T] = [M^0L^0T^-1] `	hertz `->` Hz
14	` text(Angular velocity) = text(angle)/text(time)`	`1/[T] = [ M^0L^0T^-1] `	rad/s
15	`text(Moment of inertia), I = Sigma mr^2`	`[M][L]^2 = [ML^2T^0]`	`kg-m^2`
16	`text(Angular momentum) = I omega`	`[ML^2] [T^-1] =[ML^2T^-1]`	`kg-m^2//s` or J-s
17	`text(Surface tension) = text(force)/text(length)`	`[MLT^-2]//[L] = [ML^0T^-2] `	N/m
18	`text(Intensity) = text(energy)/text(area x time )`	`([ML^2T^-2])/([L^2][T]) = [ ML^0T^-3]`	`J//m²-s -> W//m²`
19	`text(Surface energy) = text(energy)/text(area)`	`[ML^2T^-2]//[L^2] = [ML^0T^-2]`	`J//m^2`
20	`text(Spring constant), k = F/x `	`[MLT^-2]//{L] = [ML^0T^-2]`	N/m
21	`text(Planck's constant), h = E/nu = text(energy)/text(frequency) `	`[ML^2T^-2]//[T^-1] = [ML^2T^-1] `	J-s
22	`text(Coefficient of viscosity) = text(force x distance )/text( area x velocity )`	`([ML^2T^-2])/([L^2][LT^-1]) = [ML^-1T^-1]`	`Nm^-2s ` or Pa-s

A scalar quantity is one whose specification is completed with its magnitude only. Two or more than two similar scalar quantities can be added according to the ordinary rules of algebra. e.g., mass, distance, speed, energy etc.

A vector quantity is a quantity that has magnitude as well as direction. Not all physical quantities have a direction. Temperature, energy, mass, and time, for example, do not "point" in the spatial sense. We call such quantities scalars, and we deal with the by the rules of ordinary algebra.

Vector quantities can be added according to the law of parallelogram or triangle law.
A vector quantity can be represented by an arrow. The front end (arrow head) represents the direction and length of the arrow gives its magnitude.

Any vector `r` can be written as `vecr=xhati+yhatj+zhatk`
where, `hati, hatj` and `hatk` are unit vectors along the perpendicular axes OX, OY and OZ, respectively.
The magnitude of vector `r` is given by `| r |=sqrt(x^2+y^2+z^2)`

`text(The Scalar Product or Dot Product)`
The scalar product of two vectors `a` and `b` is written as `veca*vecb` and is defined to be `veca*vecb=abcosphi`
where, `phi` is the angle between the vectors `a` and `b`.
Because of the notation, `veca*vecb` is also known as the dot product and is spelled as "`veca` dot `vecb`."

`text(The Vector Product or Cross Product)`
The cross product or vector product of two vectors `veca` and `vecb` is defined as the product of the magnitude of vectors `a` and `b` and sine of the between them. `vecaxxvecb=ab sintheta hatn`
where, `vecn` is a unit vector perpendicular to `veca` and `vecb`.

Because of the notation, `vecaxx vecb` is also known as the cross product, and it is spelled as "`veca` cross `vecb`".

If the position of an object is continuously changing with respect to its surroundings with time, then it is said to be in the state of motion. A bird flying in air, a train moving on rails, a ship sailing on water, are some of the examples of motion.

The length of the actual path travelled by an object during motion in a given interval of time is called the distance travelled by the object. It is a scalar quantity, i.e. it does not depend on direction.

The shortest distance between the initial and final positions of any object during its motion is called the displacement of the object. It is a vector quantity i.e. it depends on direction.

It is defined as the total path length (i.e. actual distance covered) divided by total time taken by the object
i.e. `text(Speed)=text(Total path length)/text(Total time taken)`
So, speed is a scalar quantity. It gives no idea a out the direction of motion of the object. Hence, speed of the object can be zero or positive but never negative.

Types of speed are given below
`(i) text(Uniform Speed)` An object is said to be moving with a uniform speed, if it covers equal distances in equal intervals of time.
`(ii) text(Non-uniform Speed)` An object is said to be moving with a non-uniform or variable speed, if it covers equal distances in equal intervals of time or unequal distances in equal intervals of time.
(iii) Average Speed The ratio of the total distance travelled by the object to the total time taken is called the average speed of the object.
`text(Average Speed)=text(Total Distance Travelled)/text(Total Time Taken)=(s_1+s_2+s_3+.......)/(t_1+t_2+t_3+........)`

where `s_1, s_2, s_3......` are the distances travelled by the object with speed `v_1, v_2, v_3...` respectively.
(a) If an object travels equal distances with speeds `v_1` and `v_2`, then `text(average speed)=(2v_1v_2)/(v_1 + v_2)`
(b) If an object covers first one-third distance with speed `a`, other one-third distance with speed `b` and last one-third distance with speed `c`, then `text(average speed)=(3abc)/(ab+bc+ca)`

`(iv) text(Instantaneous Speed)` When an object is travelling with variable speed, then its speed at a given instant of time is called as instantaneous speed of the object.
`text(Instantaneous Speed)=lim_(Delta->0) (Deltas)/(Deltat)=(ds)/(dt)`

It is defined as the ratio of displacement and the corresponding time interval taken by the object, i.e.
`text(Velocity)=text(Displacement)/text(Time Interval)`
Different types of velocity are given below
`=>text(Uniform Velocity)` If an object undergoes equal displacements in equal intervals of time, then it is said to be moving with uniform velocity.
`=>text(Non-uniform Velocity)` If an object undergoes unequal displacements in equal intervals of time, then it is said to be moving with a non uniform or variable velocity.
`=>text(Average Velocity)` The ratio of the total displacement to the total time taken by an object is called the average velocity of the object.
`text(Average Velocity)=text(Total Displacement)/text(Total Time Taken)`
`=>text(Instantaneous Velocity)` When an object is travelling with variable velocity, then its velocity at given instant of time is called as instantaneous velocity.
`text(Instantaneous Velocity), v=lim_(Deltat->0)(Deltar)/(Deltat)=(dr)/(dt)`

Acceleration of an object is defined as the rate of change of velocity. It is a vector quantity having unit `m//s^2`. Acceleration can be positive, zero or negative. Positive acceleration means velocity increasing with time, zero acceleration means velocity is uniform, while negative acceleration (retardation) means velocity is decreasing with time.

`text(Average and Instantaneous Acceleration)`
Average acceleration is defined as the change in velocity (`Deltav`) divided by the time interval (`Deltat`).
Let us consider the motion of a particle. Suppose that, the particle has velocity `v_1` at `t = t_1` and at a later time `t = t_2` it has velocity `v_2`. Thus, the average acceleration during the time interval `Deltat= t_2 - t_1`, is
`a_(av)=(v_2-v_1)/(t_2-t_1)=(Deltav)/(Deltat)`

On a plot of velocity versus time, the average acceleration is the slope of the straight line connecting the points `(v_2 , t_2 )` and `( v_1, v_2)`.
If the time interval approaches zero, average acceleration is known as instantaneous acceleration.
Mathematically, `a=lim_(Delta->0) (Deltav)/(Deltat)=(dv)/(dt)`
When a body is moving with a constant acceleration, then acceleration time graph is a straight line.
Negative acceleration is known as retardation. It indicates that velocity is decreasing with respect to time.
Velocity and acceleration of an object need not be zero simultaneously.

`text(Uniform Acceleration and Non-Uniform Acceleration)`
`text(Uniform Acceleration)` An object is said to be moving with uniform acceleration, if its velocity changes by equal amounts in equal interval of time.
`text(Non-Uniform Acceleration)` An object is said to be moving with a non-uniform or variable acceleration, if its velocity changes by unequal amount in equal intervals of time.

If only one out of three coordinates specifying the position of the object changes with respect to time, then the motion is called one-dimensional motion. It is also known as rectilinear or linear motion.
e.g. (i) Motion of a train along a straight line.
(ii) Motion of freely falling objects.

Equations for one-dimensional motion also known as kinematic equations of motion.

(i) `v=u+at`
(ii) `s=ut+1/2at^2`
(iii) `v^2=u^2+2as`

where, `u=` initial velocity
`v=` velocity at time t
`s=` displacement of particle at time t
`a=` acceleration

`=>` If an object starts from rest, then `u =0`.
`=>` If an object comes to rest (i.e. it stops), then `v=0`.
`=>` If an object moves with uniform velocity, then its acceleration, `a=0`.

`text(Graphs in One-Dimensional Motion)`
Graphs can be generalized on the basis of position-time and velocity-time in one-dimensional motion with uniform velocity and uniform acceleration.

If an object is thrown upwards or falling downwards, then its motion is called motion under gravity.

`text(Equations for Upward Motion)`
(i) `v=u-g t`
(ii) `h=ut-1/2g t^2`
(iii) `v^2=u^2-2g h`

`text(Equations for Downward Motion)`
(i) `v=u+g t`
(ii) `h=ut+1/2g t^2`
(iii) `v^2=u^2+2g h`
When an object is falling freely under gravity then, `u=0`

Here, `h=` Height
`g=` Acceleration due to gravity
`u=` Initial Velocity
`v=` Final Velocity
`t=` Time

If an object is dropped vertically downwards with some height and another object is projected horizontally, then both the objects will reach the ground at same time. Velocity and acceleration of an object may be in different directions.

If only one out of three coordinates specifying the position of the object changes with respect to time, then the motion is called two-dimensional motion. e.g.
(i) Motion of a car on a circular turn.
(ii) Motion of a billiards ball.

`text(Equations for Two-Dimensional Motion)`
To deal with motion on a plane (xy) we have to break the motion along x and y-directions, then we will apply equation of motion of one-dimension separate for x and y-axes.

We considered objects moving along straight line paths, such as the x-·axis. Now, let us look at some cases in which an object moves in a plane. By this we mean that the object has a motion in both the x and the y-directions simultaneously.

Which is another way of saying, it moves in two dimensions. The motion of a particle thrown in a vertical plane, making an angle with the horizontal (`ne90^@`), is an example of two-dimensional motion. This is called projectile motion.

`text(Time of Flight)`
It is defined as the total time for which the projectile remains in air, `T=(2usin theta)/g`

`text(Maximum Height)`
It is defined as the maximum vertical distance covered by projectile, `H=(u^2sin^2theta)/(2g)`

`text(Horizontal Range)` It is defined as the maximum distance covered in horizontal distance, `R=(u^2sin2theta)/g`
`=>` Horizontal range is maximum when it is thrown at an angle of `45^@` from the horizontal `R_(max)=u^2/g`
`=>` For angle of projection `theta` and `(90-theta)` the horizontal range is same.

The circular motion differs from the linear motion in one very important aspect that in a circular motion particles move along circular track such that direction of motion changes continuously unlike in a linear motion. Therefore, circular motion is described in terms of angular displacement i.e. angle turned by the rotating body in an unit time.
When an object moves circular path with a constant speed then the motion of the object is said to be a uniform circular motion.

`text(Angular Displacement)(theta)`
It is defined as the angle turned by the particle from some reference line. Angular displacement `Delta theta` is
usually measured in radian.
Finite angular displacement `Delta theta` is a scalar but an infinitesimally small displacement: is a vector.
`text(Angular Displacement)(theta)=text(length of arc)/text(radius of circle)=(Deltas)/r`

`text(Angular Velocity)(omega)`
It is defined as the rate of change of the angular displacement of the body.
Angular velocity, `omega=lim_(Delta t->0)((Delta theta)/(Delta t))=(d theta)/(dt)`
`omega=(2pi)/T`
It is an axial vector whose direction is given by the right hand rule. Its unit is rad/s.

`text(Angular Acceleration)(alpha)`
It is the rate of change of angular velocity.
Thus, `alpha=(domega)/(dt)=(d^2theta)/(dt^2)`
Its unit is `rad//s^2`.

`text(Centripetal Acceleration)`
Acceleration acting on the object undergoing uniform circular motion is called centripetal acceleration. It always act on the object along the radius towards the centre of the circular path.
`text(Centripetal Acceleration)(a)=v^2/r=omega^2r`

`text(Velocity)(v)`
A particle in a circular motion has two types of velocities and corresponding two speeds.
(i) Linear velocity (v) or speed (v)
`v=(ds)/(dt)` and `v=|v|=|(ds)/(dt)|`
(ii) Angular velocity (`omega`) or speed (`omega`)
`omega=(d theta)/(dt)` and `omega=|omega|=|(d theta)/(dt)|`
Relation between linear speed ( v) and angular speed (`omega`) is `v=romega`
In vector form, `vecv=vecomegaxxvecr`
Here, r is the position vector of particle with respect to the centre of the circle.

`text(Centripetal Force)`
Centripetal force is that force which is required to move an object along the radius and towards centre.
`text(Centripetal force)(F)=(mv^2)/r=momega^2r`

`text(Resultant Acceleration)`
The value of resultant acceleration in non-uniform circulation is, `a=sqrt(a_R^2+a_T^2)`
where, `a_R=` Radial acceleration `= v^2/r`
`a_T=` Tangential acceleration `=alphaxxr`

Force is an external effort in the form of push or pull which
(i) generates or tends to generate motion in a body at rest,
(ii) stops or tends to stop a body in motion,
(iii) increases or decreases the magnitude of velocity of the moving body,
(iv) changes the direction of motion of the body.
There are two types of force
(i) `text(Balanced Forces)` If there are many forces acting on an object but resultant of all of them is zero, then the forces arc called balanced forces.
(ii) `text(Unbalanced Forces)` If the resultant of all the forces acting on an object is not zero, then the forces are called unbalanced forces.

If two forces are acting on a point simultaneously, whose magnitudes and directions can be shown by two adjacent sides of parallelogram, then the magnitude and direction of resultant force will be shown by the diagonal which passes through the point of intersection of those sides.
Resultant of these two forces P and Q is `R=sqrt(P^2+Q^2+2PQcosalpha)`
where, `alpha` is the angle between the sides of parallelogram.
`tanbeta=(Qsin theta)/(P+Qcostheta)`
where, `beta` is the angle made by the resultant with P.

It is an inherent property of all bodies, by virtue of which they cannot change by themselves their state of rest or of uniform motion along a straight line.
As inertia of a body is measured by the mass of the body. So, heavier the body, greater is the force required to change its state.
There are three types of inertia
(i) `text(Inertia of Rest)` If an object resists the change in its state of rest, its inertia is called inertia of rest.
(ii) `text(Inertia of Motion)` If an object resists the change in its state of motion, its inertia is called inertia of motion.
(iii) `text(Inertia of Direction)` If an object resists the change in direction of its motion, its inertia is called inertia of direction.

The linear momentum of a body is defined ash product of mass and velocity of the body, i.e.
`p=mv`
Momentum is a vector quantity.
A body at rest cannot possess linear momentum and a moving body always possesses linear momentum.

If the net force `sumF` exerted on an object is zero, then the object continues in its original state of motion (or rest). That is, if `sumF=0`, an object at rest remains at rest or object moving with constant velocity. This is Newton's first law.
This law is also known as law of inertia.

`text(Applications of Newton's First Law of Motion)`
(i) The passengers in a bus falls backward when it starts suddenly. This is because the sudden start of the bus brings motion to the bus as well as to our feet in contact with the floor of the bus but the rest portion of our body opposes this motion because of inertia, so they fall backwards.
(ii) When a carpet or a blanket is beaten with a stick, then the dust particles separate out from it.

The acceleration of an object is directly proportional to the net force acting on it and is inversely proportional to its mass.
`a=(sumF)/m`
This is the Newton's second law.
According to the second law of motion, force `Fproptext(change in momentum)/text(time)`
`F=K(p_2-p_1)/t=K (m(v-u))/t=Kma`, K = constant of proportionality.
Its value is one in SI and CGS system.

`text(Application of Newton's Second Law of Motion)`
(i) A cricket player (or fielder) moves his hands backward on catching a fast cricket ball.
(ii) During athletics meet, a high jumping athlete is provided either a cushion or a heap of sand on the ground to fall upon.

According to the principle of conservation of linear momentum, if there is no external force acting on a system, then total momentum of the system remains constant.
According to second law of motion, `F=(dp)/(dt)`
If no force is acting, then `F=0`
So, `(dp)/(dt)=0 => p=` constant
`m_1v_1=m_2v_2=` constant

If two objects interact, the force `F_(12)` which the object 1 exerts on object 2 is equal in magnitude but opposite in direction to the force `F_(21)` which the object 2 exerts on object 1. This is Newton's third law, i.e. `F_(12) = - F_(21)`

`text(Applications of Newton's Third law of Motion)`
(i) `text(Walking of a person)` A person is able to walk because of the Newton's third law of motion. During walking, a person pushes the ground in backward direction and in the reaction the ground also pushes the person with equal magnitude of force but in opposite direction. This enables him to move in forward direction against the push.
(ii) `text(Recoil of gun)` When bullet is fired from a gun, the bullet also pushes the gun in opposite direction with equal magnitude of force. This results gunman feeling a backward push, i.e., recoil force from the butt of gun.
(iii) `text(Propulsion of a boat in forward direction)` Sailor pushes water with oar in backward direction resulting water pushes the oar in forward direction.
Consequently, the boat is pushed in forward direction. Force applied by oar and water are of equal magnitude but in opposite directions.
(iv) `text(Rocket propulsion)` The propulsion of rocket is based on the principle of action and reaction. The rapid burning of fuel produces hot gases which rush out from the nozzle at the rear end at a very high speed. The equal and opposite reaction force moves the rocket upward at a great speed.