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

MACHINES

MACHINE
A machine is a device by means of which work can be performed easily or in a convenient manner. A machine can be used :  To lift heavy loads by applying little force.  To enlarge magnitude of force  To increase rate of work done  To change the direction of force Example of simple machines are : Lever, pulley, inclined plane, wedge, screw etc.
EFFORT OR POWER
The power directly applied to a machine to lift a load is called Effort or Power. It is denoted by ‘P’.
LOAD OR WEIGHT
The weight lifted by a machine is called Load. It is denoted by ‘W’.
MECHANICAL ADVANTAGE
The ratio of weight (load) lifted by a machine to the force(effort) applied on a machine is called mechanical advantage of the machine. Greater the value of mechanical advantage of a machine, more easier is the work done. Mathematically,
M.A = load/effort
OR
M.A = W/P
UNIT:
It has no unit.
INPUT
Amount of work done on a machine by a given effort (force) is called input of a machine.
Input = effort x distance through which effort acts OR
input = P x d
OUTPUT
Amount of work done by a machine on the load (weight) is called output of the machine.
Output = load x distance covered by the load OR
Output = W x D
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EFFICIENCY
The ratio of output of a machine to the input of machine is called its efficiency.
h = output/input h = (W x D)/(P x d) Efficiency in %:                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         h = (W x D)/(P x d )x100
UNIT:
It has no unit.
IDEAL MACHINE
An ideal machine is a hypothetical machine whose output is equal to its input. For an ideal machine
output = input
Efficiency of an ideal machine is 100% because there is no loss of energy in an ideal machine due to friction or any other means that can waste useful energy.
M.A of an ideal machine is d / h.
LEVER
Lever is a simple machine which is used to lift heavy bodies or heavy load in a very easy way. Lever consists of a rigid bar capable to rotate about a fixed axis called fulcrum. Effort is applied at one end of the bar and weight can be lifted from the other end.
TYPES OF LEVER
There are three kinds of lever depending upon the positions of load , effort and fulcrum.
FIRST KIND OF LEVER
In the first kind of lever, the fulcrum F lies between effort (P) and load (W).
Example: common balance, seesaw, scissors, handle of hand pump.
SECOND KIND OF LEVER
In the second kind of lever, load (W) lies between effort (P) and fulcrum (F).
Example: door, nutcracker, punching machine.
THIRD KIND OF LEVER
In the third kind of lever, effort (P) lies between load (W) and fulcrum (F). Example: forceps, jaws, human forearm, firetong.

STATES OF EQUILIBRIUM

States of equilibrium
There are three states of equilibrium:
Stable equilibrium    Unstable equilibrium    Neutral equilibrium
Stable equilibrium
When the center of gravity of a body lies below point of suspension or support, the body is said to be in    STABLE EQUILIBRIUM. For example a book lying on a table is in stable equilibrium.
Explanation
A book lying on a horizontal surface is an example of stable equilibrium. If the book is lifted from one edge    and then allowed to fall, it will come back to its original position.    Other examples of stable equilibrium are bodies lying on the floor such as chair, table etc.
Reason of stability
When the book is lifted its center of gravity is raised . The line of action of weight passes through the    base of the book. A torque due to weight of the book brings it back to the original position.
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Unstable equilibrium
When the center of gravity of a body lies above the point of suspension or support, the body is said to    be in unstable equilibrium
Example
pencil standing on its point or a stick in vertically standing position.
Explanation:    If thin rod standing vertically is slightly disturbed from its position it will not come back to its original    position. This type of equilibrium is called unstable equilibrium, other example of unstable equilibrium are    vertically standing cylinder and funnel etc.
Reason of instability
when the rod is slightly disturbed its center of gravity is lowered . The line of action of its weight lies    outside the base of rod. The torque due to weight of the rod toppled it down.
Neutral equilibrium
When the center of gravity of a body lies at the point of suspension or support, the body is said to be in    neutral equilibrium. Example: rolling ball.
Explanation
If a ball is pushed slightly to roll, it will neither come back to its original nor it will roll forward rather it will    remain at rest. This type of equilibrium is called NEUTRAL EQUILIBRIUM.
Reason of neutral equilibrium
If the ball is rolled, its center of gravity is neither raised nor lowered. This means that its center of gravity    is at the same height as before.

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

DIFFERENCE BETWEEN MASS AND WEIGHT

Mass		Weight
(1) The quantity of matter in a body is          called its mass.		(1) Weight is the force by which the earth          attracts a body towards its center.
(2) Mass is a scalar quantity.		(2) Weight is a vector quantity and is always          directed towards the center of the earth.
(3) Mass of a body is always constant          every where in the universe.		(3) Weight of a body vary place to place and          become zero on the center of earth and far          away from the surface of earth.
(4) Mass of a moving body is m=F/a.		(4) Weight of a body is W = mg.
(5) Mass can be determine by an          ordinary balance.		(5) Weight of a body is measured by spring          balance.
(6) Unit of mass in S.I system is          KILOGRAM ( kg).		(6) Unit of weight in S.I system is NEWTON (N).
FRICTION
When a body slides over the surface of another body, an opposing force is set up between them to resist    the motion. The force which opposes the motion is called friction OR Force of Friction.
Force of friction tends to decelerate a body and always acts in the opposite direction of motion.
CATEGORIES OF FRICTION    (1) Contact friction    (2) Fluid friction
LIMITING FRICTION
When an external force is applied against the force of friction, the force of friction also increases by the    same amount. Therefore, It adjusts itself in such a way that it is equal and opposite to the external    force. It has a maximum value just before the motion starts. So friction is a self-adjusting force. The    maximum force of friction that stops the body from moving is called LIMITING FRICTION.    It is denoted by Fs.    LIMITING FRICTION is directly proportional to the surface reaction.
Limiting friction Fs is:
Where R = normal reaction  but R = W and R = mg
Where = constant known as coefficient of friction.
COEFFICIENT OF FRICTION
Coefficient of friction is the ratio of LIMITING FRICTION to the NORMAL REACTION.
Coefficient of friction is constant for a given pair of surfaces but different for different pairs
Unit of :    Since it is a ratio of two similar quantities, therefore it has no unit as shown.
ROLLING FRICTION		www.citycollegiate.com
When a body rolls over a surface, the force of friction is called ROLLING FRICTION. When a sphere rolls    over a surface it experiences an opposing force called ROLLING FRICTION.    Rolling friction is much less than the sliding friction because in case of rolling contact area of two surfaces    is very small as compared to sliding.

DISTRIBUTIVE LAW FOR DOT PRODUCT

DISTRIBUTIVE LAW FOR DOT PRODUCT
According to distributive law for dot product:
	PROOF
	Consider three vectors , and .Here we will use geometric interpretation of dot product by drawing projection as shown below. First we obtain the sum of vectors  and by head to tail rule then we draw projection  and from the terminal point of vector respectively onto the direction of . The dot product  is equal to the projection of vector  onto the direction of  multiplied by the magnitude of. i.e.