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In physics we deal with two type of physical quantity one is scalar and other is vector. Each scalar quantity has a magnitude and a unit.
Example of scalar quantities: mass, speed, distance etc.
Scalar quantities can be added, subtracted and multiplied by simple laws of algebra.
Vector are the physical quantities having magnitude as well as specified direction.
Example of vector quantity: Displacement, velocity, acceleration, force etc.
Knowledge of direction
Representation of vector: Geometrically, the vector is represented by a line with an arrow- indicating the direction of vector as
Mathematically, vector is represented by A. Sometimes it is represented by bold letter A. Thus, the arrow in abow figure represents a vector
in xy-plane making an angle (3 with x-axis.
A representation of vector will be complete if it gives us direction and magnitude.
Symbolic form: v, a. F. s used to separate a vector quantity from scalar quantities (u. i, m) Graphical form: A vector is represented by a directed straight line.
having the magnitude and direction of the quantity represented by it.
Angle between two Vectors (θ) Angle between two vectors means smaller of the two angles between the vectors When they are placed tail to tail by displacing either of the vectors parallel to itself (i. e 0 ≤ θ ≤π).
Negative of Vector
It implies vector of same magnitude but opposite in direction.
Equality of Vectors.
Vectors basing equal magnitude and same direction are called equal vectors
Any two vectors are co-linear then one can be express in the term of other.
a = λb (where λ is a constant)
Co-initial vector:If two or more vector start from same point then they called co-initial vector.
Here, A,B, C, D are co-initial.
Three (or more) vectors are called coplanar vectors if they lie in the same plane or are parallel to the same plane. Two (free) vectors are always coplanar.
Multiplication and division of a vector by a scalar
Multiplying a vector A with a positive number λ gives a vector (B= λ A) whose magnitude become λ, times but the direction is the same as that of A. Multiplying a vector A by a negative number λ gives a vector B whose direction is opposite to the direction of A and whose magnitude is - λ. times |A|.
The division of vector A by a non-zero scalar m is defined as multiplication of
At here A and B are co-linear vector
REST AND MOTION:
Motion is broadly classified into 3 categories.
1.1 Rectilinear or 1-D Motion
When a particle is moving along a straight line, then its motion is a rectilinear motion.
Parameters of rectilinear motion or translator motion or plane motion:
(B) Position or location - It is defined with respect to some reference point (origin) of given frame of reference.
Consider particle which moves from location r1 (at time t1)
To location r2 (at time t2) as shown in the figure below, following path ACB.
The length of the actual path traversed by the particle is termed as its distance.
Distance = length of path ACB.
The change in position vector of the particle for a given time interval is known as its displacement.
For a moving particle in a given interval of time
(E)Average speed and average velocity:
Average speed and average velocity are always defined for a time interval.
Instantaneous speed and instantaneous velocity
Instantaneous speed is also defined exactly like average speed i.e. it is equal to the ratio of total distance and time interval, but with one qualification that time interval is extremely (infinitesimally) small.
The instantaneous speed is the speed at a particular instant of time and may have entirely different value than that of average speed. Mathematically
Instantaneous velocity is defined exactly like speed. It is equal to the ratio of total displacement and time interval, but with one qualification that time interval is extremely (infinitesimally) small. Thus instantaneous velocity can be termed as the average velocity at a particular instant of time when At tend to zero and may have entirely different value that of average velocity : Mathematically.
When a particle moves in a plane such that its distance from a fixed (or moving) point remains constant then its motion is called as the circular motion with respect to that fixed (or moving) point. That fixed point is called centre and the distance between fixed point and particle is called radius.
KINEMATICS OF CIRCULAR MOTION:
2.1 Variables of Motion:
(a) Angular Position:
The angle made by the position vector with given line (reference line) is called angular position Circular motion is a two dimensional motion or motion in a plane. Suppose a particle P is moving in a circle of radius r and centre O. The position of the particle P at a given instant may be described by the angle θ between OP and OX. This angle θ is called the angular position of the particle. As the particle moves on the circle its angular position θ change. Suppose the point rotates an angle ∆θ in time ∆t.
Angle rotated by a position vector of the moving particle in a given time interval with some reference line is called its angular displacement.
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