E1.5 Work - Energy Theorem:

Net work on an object causes a change  in the kinetic energy. The work done in a body due to a force is equal to the change in its kinetic energy. 

i.e. Work done = Change in Kinetic Energy

Consider a body of mass $M$, moving with a initial velocity $u$  when a constant force $F$ is acts on it. Let final velocity of the body becomes $v$ after covering a distance $d$ in a direction of force. Then, work done by the force is given by

$W = F.d$ .......... (i)

From the Newton's law of Motion:

$F = ma$ .......... (ii)

From equation (i) and (ii)

$F = ma.d$ .......... (iii)

And from the Kinematics, we know the formula:

$v^2 = u^2 + 2ad$

$d = \frac{v^2 - u^2}{2a}$ .......... (iv)

From the equation (i), (iii) and (iv), we get

$W = ma\;(\frac{v^2 - u^2}{2a})$

      $= m\;(\frac{v^2}{2} - \frac{u^2}{2})$

     $= \frac{1}{2}\,mv^2 - \frac{1}{2}\,mu^2 $

      = Final K.E - Initial K.E.

∴ Work done = Change in Kinetic Energy

So, the work done on moving the body by the force is equal to the increase in its kinetic energy.

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E1.5 Work, Energy and Power (Work):

In general, the work done by a force is equal to the product of the force and the displacement of its point of application in the direction of the force. 

Work (symbol: W): is defined as transfer of energy by force from one object to another object. If one object transfers energy to a another object , then said that the first object does work on the second object.

Suppose, a force is applied on an object and object is displaced from its position due to the application of the force. It is said to be work done by the force. There are three major component to work: 'Force, Displacement and Cause'. In order to force to qualify as having done work on an object, there must be a displacement and the force must cause the displacement (the physics classroom).

Mathematically,

Work is defined as the product of force and displacement of an object in the direction of force.

i.e. $W = \vec{F}.\vec{d}$ .......... (i)

Where; W  =  Work done; 

              F  =  Force; 

              d  =  Displacement.

If $\theta$ is the angle between $\vec{F}$ and $\vec{d}$ then, 

Work done by the force (W) = Force * distance moved in direction of force.

$W = F . d\; Cos \theta$ .......... (ii)

If the force and and displacement are in the same direction, then $\theta = 0^{0} \;\;(i.e.\;\; Cos0^{0} = 1)$ From the equation (ii), we have

∴ $W = Fd$ .......... (iii)

Note: Here, no work is done by $F\;Sin\theta$ component because displacement in the direction of this component is zero.

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» Unit of work is joule ($J$), in SI system. It's dimensional is [$M\;L^2\; T^{-2}$].

Relation between Joule and erg is,

$1\;J = 1\;N * 1\;m = 10^5\; dyne * 100\;cm = 10^7\;erg $

Work is a scalar quantity, it has no property of direction but only magnitude. When the force is one Newton and the distance moved is one meter, then the work done is one Joule.

Another Example: A force of 50 N moving through a distance of 10 m does 50 * 10 = 500 Joule of work. (This is also a measure of the energy transferred to the object).

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» The angle measure is defined as the angle between the force and the displacement.

Case (i): The force vector and the displacement vector are in the same direction. That is, the angle between $F$ and $d$ is $0$ degrees. A force acts rightward upon an object as it is displaced rightward.

Case (ii): The force vector and the displacement vector are in opposite direction. That is, the angle between $F$ and $d$ is $180$ degrees. A force acts leftward upon an object that is displaced rightward.

Case (iii): The force vector and the displacement vector are at right angles to each other i.e. angle between $F$ and $d$ is $90$ degrees. If the force is perpendicular to the direction of displacement , work done is zero. 

Example: A waiter who carried a tray full of meals above his hand. The force supplied by the waiter on the tray is an upward and the displacement of the tray is a horizontal with a waiter speed. In this case, the angle between the force and displacement is $90$ degrees. If we calculated the work done by waiter on the tray, the result would be zero.

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E1.4 Force and Inertia:

Galileo Galilei (1564 - 1642): a premier scientist in the $17^{th}$ century. He developed the concept of inertia. The law of inertia is the basis of the physics of the $17^{th}$ century. This law is also true according to modern physics.

"The natural tendency of an object to remain at rest unless acted upon by an external force." 

The concepts of inertial in Galileo's writing would later come to be modified by Isaac Newton as the first of his Laws of Motion.

Laws of Inertia: 

Galileo thought that, a moving objects eventually stop because of a force called friction. In experiment, using a pair of inclined planes facing each other, as shown in figure. Let us imagined this motion:

Galileo imagined a ball roll down from a left inclined plane and roll up to same height of the right inclined plane. If smoother plane were used, the ball would roll up closure to the original height of the opposite plane. If the rolling ball did not reach the original height of the opposite plane(i.e. difference between initial and final height), there presence some external force called Friction. Galileo conclude that: "If friction could be entirely eliminated then, the ball would reach exactly the same height."

Galileo demonstrated that: if the slope of right inclined plane is slightly reduced, then ball would roll for an even longer time on the right side plane before coming to a stop. Again if the slope of  right inclined plane were reduced to $0 - degree$ then, ..... a ball in motion would continue in its state of motion..... moving at a constant speed in a straight line. This property of matter is called inertia.

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Inertia is the tendency of an object to continue in its original state. A body at rest moves only when a force acts on it. Velocity of a uniformly moving body will change only when a force acts on it. 

1. Inertia of rest: It is the tendency of a body to remain in its state of rest.

Example: 

a) Passengers in a bus fall backward if the bus suddenly starts to move due to inertia of rest. 

b) Place a coin on a paper. Pull the paper out suddenly. the coin remains in its original position due to inertia of rest.

2. Inertia of motion: It is the tendency of a body to remain in its uniform motion.

Example:

a) Passengers in a bus fall forward when the bus suddenly stops. 

b) To jump a long jump an athlete has to run a long distance before taking a jump. 

Inertial Mass: 

Inertial frames:

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E1.4 Dynamics (Newton's Laws of Motion):

"I do not know what I may appear to the world; but to myself I seem to have been only like a boy playing on the seashore, and diverting myself in now and then finding a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me."

~ Sir Isaac Newton

Sir Isaac Newton (1642-1726): A Mathematician, Astronomer and Physicist was born at Wollsthrope in England. He worked in many area of Mathematics and Physics. 

He developed the theories of gravitation in 1666, when he was only 23 years old. After some year later, he presented his three laws of motion in the book: "Principa Mathematica Philosophiae Naturalis" in 1687. 

Click here for: Vedic Laws of Motion

#1. Newton's $1^{st}$ Law:

"A body remains in the state of rest or in the state of uniform velocity if no net force is acting on it." 

It is also known as law of Inertia. 

(If the net force is zero: Object at rest, stay at rest. And Object in motion, continue to move). 

This means that, there is a natural tendency of objects to keep on doing what they're doing. There is no change in the state (rest or motion) without an external force acting. That is, a force is an agent that produces a change in state of a body. 

 

#2. Newton's $2^{nd}$ Law:

"The rate of change of momentum of a body with respect to time is directly proportional to the net external force acting on the body."

However, Newton's second law gives us an exact relationship between force, mass and acceleration. Mathematically,

$F_{net} = ma$

i.e.  net force on object = mass of object * acceleration

If we exert the same force on two objects of different mass, we'll get different acceleration (change in motion).The heavier objects require more force to move than the lighter objects.

Note: In equation, $F = ma$; if net force acting on a mass is zero then, $a=0$.

This means that, if the net force acting on a body is zero, it moves with constant velocity or it will be rest. Which is the statement of the Newton's $1^{st}$ law. 

#3. Newton's  $3^{rd}$ Law:

"To every action there is always an equal and opposite reaction."

This means that, for every action there is equal and opposite reaction. It dose not matter which force we call action and which we call reaction.

All forces always occur in pair. For example: During the simplest act of walking, we push against the road surface with our shoes and road pushes back with an equal force but in opposite direction. It is the force the road exerts on us that causes us to move forward. 

Similarly, at the swimming time a swimmer moves through the water only because he pushes water backward and the water pushes the swimmer forwards.

Again, Let's understand how a rocket works. The rocket's action is to push down on the ground with the force of its powerful engines, and the reaction is that the ground pushes the rocket upwards with an equal force.

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