SIMPLE PENDULUM

SIMPLE PENDULUM
	simple pendulum consists of a heavy mass particle suspended by a light, flexible and in-extensible string.
	MOTION OF THE BOB OF SIMPLE PENDULUM
	The motion of the bob of simple pendulum simple harmonic motion if it is given small displacement. In order to prove this fact consider a simple pendulum having a bob of mass 'm' and the length of pendulum is 'l'. Assuming that the mass of the string os pendulum is negligible. When the pendulum is at rest at position 'A', the only force acting is its weight and tension in the string. When it is displaced from its mean position to another new position say 'B' and released, it vibrates to and fro around its mean position.
	Suppose that at this instant the bob is at point 'B' as shown below :
	FORCES ACTING ON THE BOB
	1. Weight of the bob (W) acting vertically downward. 2. Tension in the string (T) acting along the string.
	The weight of the bob can be resolved into two rectangular components:
	a. Wcosq along the string. b. Wsinq perpendicular to string.
	Since there is no motion along the string, therefore, the component Wcosq must balance tension (T) i.e.                                                                      Wcosq = T
	This shows that only Wsinq is the net force which is responsible for the acceleration in the bob of pendulum.
	According to Newton's second law of motion Wsinq will be equal to m x a
	i.e.                                                                      Wsinq = m a
	Since Wsinq is towards the mean position, therefore, it must have a negative sign. i.e.                                                                     m a = -  Wsinq
	But W = mg
	m a = -  mgsinq
	a = -  gsinq
	In our assumption q is very small because displacement is small, in this condition we can take sinq = q
	Hence                                                                     a = -  gq ----------- (1)
	If x be the linear displacement of the bob from its mean position, then from figure, the length of arc AB is nearly equal to x
	From elementary geometry we know that:
	Where s= x, r = l
	Putting the value of q in equation (1)
	As the acceleration of the bob of simple pendulum is directly proportional to displacement and is directed towards the mean position, therefore the motion of the bob is simple harmonic when it is given a small displacement

Momentum- Law of conservation of Momentum

MOMENTUM
Quantity of motion of a body is referred to as "MOMENTUM".
Definition
Momentum of a moving body defined as :
"the product of mass and velocity of a body is called MOMENTUM."
Mathematically
Momentum = mass x velocity
It is a vector quantity. Momentum is always directed in the direction of velocity.    The unit of momentum is in S.I system kg .m/s or NS.    Momentum depends upon mass and velocity of body.
LAW OF CONSERVATION OF MOMENTUM.
The law of conservation of momentum states that:
"when some bodies constituting an isolated system act upon one another, the total momentum of the system remains constant." OR "the total momentum of an isolated system of interacting bodies remains constant." OR "Total momentum of an isolated system before collision is always equal to total momentum after collision."
Consider an isolated system of two bodies 'A' and 'B' as shown. The masses of bodies are ma and mb and
MATHEMATICAL REPRESENTATION
Consider two bodies of mass m1 and m2 moving initially with velocities u1 and u2.
Total momentum before collision = m1u1 + m2u2
Let after collision their velocities become v1 and v2.
Total momentum after collision = m1v1 + m2v2
According to the law of conservation of momentum
m1u1 + m2u2 = m1v1 + m2v2

MULTIPLICATION & DIVISION OF VECTOR BY A NUMBER (SCALAR)

MULTIPLICATION OF A VECTOR BY A SCALAR
When a vector is multiplied by a positive number (for example 2, 3 ,5, 60 unit etc.) or a scalar only its magnitude is changed but its direction remains the same as that of the original vector. If however a vector is multiplied by a negative number (for example -2, -3 ,-5, -60 unit etc.) or a scalar not only its magnitude is changed but its direction also reversed.
The product of a vector by a scalar quantity (m) follows the following rules:
	(m)  = (m) which is called commutative law of multiplication.
	m(n) = (mn) which is called associative law of multiplication .
	(m + n) = m+ n which is called distributive law of multiplication .
	DIVISION OF A VECTOR BY A SCALAR
	The division of a vector by a scalar number (n) involves the multiplication of the vector by the reciprocal of the number (n) which generates a new vector.
	Let n represents a number or scalar and m is its reciprocal then the new vector  is given by :
	where m = 1/n
	and its magnitude is given by:
	The direction of  is same as that of if (n) is a positive number.  The direction of  is opposite as that of if (n) is a negative number.

ADDITION OF VECTORS

ADDITION OF VECTORS4

PARALLELOGRAM LAW OF VECTOR ADDITION
Acccording to the parallelogram law of vector addition:
"If two vector quantities are represented by two adjacent sides or a parallelogram then the diagonal of parallelogram will be equal to the resultant of these two vectors."
	EXPLANATION
	Consider two vectors . Let the vectors have the following orientation
	parallelogram of these vectors is :
	According to parallelogram law:
	MAGNITUDE OF RESULTANT VECTOR
	Magintude or resultant vector can be determined by using either sine law or cosine law.