GRAVITATION

Gravitation and gravity

• Gravitation is the force of attraction between any two bodies 
• Gravity is the force of attraction between a body and the Earth. It pulls all the objects towards the earth. It holds the stars together.

 Gravitational force- Gravitational force is the force with which earth attracts the other body.

Mass and Weight 

• Mass is the quantity of matter contained in the body.
• Mass is a scalar quantity which has only magnitude but no direction.
• Mass does not change from place to place, it is always remains constant and SI unit of mass is kilogram (kg) and Mass of a body can never be zero
• Weight of the body is the force with which Earth attracts the body. It depends on Mass(m) and gravity (g). Weight of the object becomes zero if g is zero. The SI unit of weight is Newton. The weight of a body is maximum at the pole and minimum at the equator of the earth.

The Law of Gravitation 

According to the law of gravitation, F is directly proportional to the product of two masses of the objects and inversely proportional to the square of the distance between them. 

If M and m are the masses, F is the gravitational force and r is the distance between two objects then'

F ∝  M x m   and F ∝ 1/r²

Therefore, F ∝ Mm/r².

Or  F = G Mm/r²

The value of G remains constant at all the places in the universe and so it a called universal constant. The value of G does not depend on the medium.

• Hence, if the mass of one object is doubled, then the gravitational force will also get doubled, If the masses of both the objects are doubled, the product of masses becomes four times and hence the gravitational force will also become four times the original value.
• If the distance between the objects is reduced to half, then the gravitational force would become four times and if the distance is increased to double, the the gravitational force would become one fourth.

Unit of gravitational constant G is Nm²kg^-2

Importance of The Universal Law of Gravitation

• It binds all the objects and us to the earth.
• It causes tides in seas on earth.
• It causes the motion of the moon around the earth.
• It causes the motion of planets around the Sun.

Factors on which, the strength of gravitational force depends

• Masses of the objects
• Distance between the objects 

Acceleration due to gravity

If an object is moving under the influence of only gravitational force, near the earth the gravitational force produces a uniform acceleration in the objects, which is called acceleration of free fall or acceleration due to gravity. It is denoted by g and its value is 9.8 m/s².

The earth is not a perfect sphere. As the radius of the earth increases from the pole to the equator, the value of g becomes smaller at the equator and greater at the pole.

To calculate the value of 'g'

• According to the Newtons third law of motion, the force of attraction between the earth and the earth and body is given by F = GMm/r²
  Where m = mass of the object, M = mass of the earth, r = distance between earth and object and G =universal constant.
• And according to Newton's second law: F = mg.  (where g is the acceleration due to gravity) Therefore, mg = GMm/r²    and g = GM/r². 
• If the body is located on the earth's surface of the earth then r = R Therefore g = GM/R². This expression gives acceleration due to gravity at the surface of the earth where the value of g is 9.8 m/s²

Factors Affecting the Value of g

• As the radius of the earth increases from the poles to the equator, the value of g increases and greater at the poles than at the equator.
• As the height increases, the value of g decreases.

Motion of Objects Under the Influence of Gravitational Force of the Earth

If an object is falling towards earth with initial velocity u and gravitational acceleration g, and final velocity v after covering the height h in time t, then the three equations of motion can be represented as:

•  v = u + ght
• h = ut + ½gt²
• v² = u² + 2gh

The value of g is  positive when the object is moving towards earth and negative when the object is thrown in opposite direction of the earth.

Weight of the object on the moon

The weight of an object on the moon is one-sixth the weight of the object on the earth as gravitational force of moon is one sixth of earth.

FORCE AND LAW OF MOTION

Force 

Force is pushing or pulling that can change the state of rest or the state of motion of an object.

• Force can change the direction of motion
• Force can change the shape and size of the object
• Force can change the speed of motion or magnitude of the velocity
• Force can bring a motion in a restin object

The expression of force is F = ma, where m = mass of the object and a = acceleration

Force It is a vector quantity and the S.I unit is Newton (N) and the CGS unit is dyne.

Types of force

Contact force - A force exerted on a body when it is in contact with the body is called contact force. Example - Frictional force, mechanical, and muscular force.

Non-contact force - A force exerted on a body without any contact with two bodies is called contact force. Example - Gravitational force, magnetic force, and an electrostatic force.

Frictional force - Frictional force is the force caused by the relative motion of two surfaces that come in contact with each other.

Resultant force - The resultant force is a net force acting on a body along with its direction. When a body at rest or moving with a uniform velocity, the net force has to be zero.

1-Newton force

1-Newton force is defined as the amount of force that produces an acceleration of 1 m/s² in an object of 1 kg mass.  

The balanced and unbalanced force

Balanced force - The net effect produced by a number of forces on a body is zero is called balanced force

Unbalanced force -  The net effect produced by a number of forces on a body is non zero and it is called unbalanced force. In an unbalanced force, two opposite forces acting on a body and the body moves in the direction of greater force. 

Momentum 

The momentum of an object is the product of its mass and velocity. 

• Its S.I. unit is kg m/s
• momentum has both direction and magnitude.
• it is a vector quantity

Conservation of momentum

If the external force on a system is zero, and the sum of the momentum of two objects before the collision is equal to the sum of momentum after the collision, then it is known as conservation of momentum.

Suppose A and B are two balls, they have mass mA and initial velocities uA and uB . When the two bodies collide their final velocities changes to vA and vB 

The force is exerted by ball A is FAB and by the ball B is FBA.  

According to the conservation of momentum

 muA+ muB= mAvA + mBvB

Linear momentum

The product of the mass and velocity of a moving body is called linear momentum.
Momentum  p = mv. (where = momentum, m = mass and v = velocity)

S.I unit of momentum is Kg m/s.

Prove the principle of conservation of linear momentum for two bodies moving in the same direction and coiling.

Suppose two balls A and B of masses m_A and m_B are moving in the same direction along a straight line at different initial velocities u_A and u_B, respectively. And there are no other external unbalanced forces acting on them.
Let u_A  is greater than u_B . The two balls collide with each other. During collision  ball 'A' exerts a force F_AB on ball 'B' and the ball 'B' exerts a force F_BA on ball 'A'.

Suppose v_A and v_B are the velocities of the two balls 'A' and 'B' after the collision, respectively. 

The momentum of ball 'A' before the collision is m_Au_A 

The momentum of ball 'B' after the collision is m_Av_A,.
The rate of change of its momentum during the collision will be
 = (m_A v_A- m_Au_{A) /t 
 =}  m_A( v_{A -}  u_{A) / t}
Similarly, the rate of change of momentum of ball B (= F_BA or reaction) during the collision will be 
= m_B ( v_B - u_{B) /} t .
According to the third law of motion,
Therefore, F_AB = – F_BA    
m_A( v_{A -}  u_{A) / t} = m_B ( v_B - u_{B) /} t 
or m_Av_{A -}  m_Au_A  = m_Bv_B - m_Bu_B
or m_Au_A+ m_Bu_B = m_Av_A + m_Bv_A 
This proves the total momentum of the two balls remains unchanged or conserved provided no other external force acts.

Inertia

All objects resist a change in their state of motion. In a qualitative way, the tendency of undisturbed objects to stay at rest or to keep moving with the same velocity is called inertia.
There are two types of inertia: The motion of inertia and the rest of inertia.

The tendency of undisturbed objects to stay at rest or to keep moving with the same velocity is called inertia. The physical quantity of inertia is mass and the S.I unit is Kg.

The relation between inertia and mass

The property of inertia is because of the mass of the body. The greater the mass, the greater is the inertia of the body. 
Steel has the highest inertia because the mass of steel is more than the aluminum and wooden solid because the greater the mass, the greater is the inertia of the body so.

Newton's first law that is known as the law of inertia

According to Newton's first law of motion, if a body is in rest, it will remain in the state of rest, and if it is in a state of motion, it will remain moving in the same direction with uniform velocity unless an external force is applied on it.
It is known as the law of inertia because this property of an object by virtue of which it neither changes its present state nor it tends to change the present state, is called inertia.

When a carpet is beaten with a stick it releases dust, why?

When a carpet is beaten, due to the inertia of rest, the dust remains in the state of rest and the carpet comes in motion and leaves the carpet and falls down.

Balanced and unbalanced force.

• When two equal forces acting on a body in the opposite direction, do not change the state of rest or motion of the object. These forces are called balanced forces. 
• When two opposite forces of different magnitudes acting on a body, the body starts to move in the direction of the greater force. These forces are called the unbalanced force acts in the direction the body moves. An unbalanced force acting on an object brings it in motion.

Differences between mass and weight

• Mass of a body is a measure of the quantity of matter contained in the body where weight is the force by which the earth attracts the body.
• Mass is a scalar quantity, weight is a vector quantity.
• It is constant for the body and does not change by changing the place of the body, weight is not constant but varies from place to place. 

Factors on which inertia depends

The factor on which inertia depends directly, are mass and acceleration.
The greater the mass, the greater is the inertia of the body.
More the mass of a body more difficult it is to move the body from rest or to stop the body if it is initially moving. 
Similarly the greater the acceleration, the greater is the inertia of the body.

Newton's second law of motion.

The first law of motion indicates that when an unbalanced external force acts on an object, its velocity changes, that is the object gets an acceleration and the acceleration of an object depends on the force applied to it. Newton's second law of motion gives the quantitative value of force.
Therefore F = ma.

Newton's second law in terms of momentum.
According to Newton's second law of motion, the rate of change of momentum of a body is directly proportional to the force applied on it and the change in momentum is in the direction in which the force is applied.

Newton's first law in terms of the second law. 
 The first law of motion can be mathematically stated from the mathematical expression for the second law of motion.
F = ma or F = m(v − u)/ t 
or F= (mv – mu)/t .That is when F = 0, v = u for whatever time, t is taken. This means that the object will continue moving with uniform velocity, u throughout the time, t. If u is zero then v will also be zero. That is, the object will remain at rest. This is newtons second law.

How karate player can break a pile of tiles in a single blow with his hands. 
 Karate player moves his hand with high speed due to which momentum force increases and strikes the pile with his hand very fast. Thus the entire momentum of the fast-moving hand is reduced to zero in a very short time, due to this, the rate of change of momentum of the ball will be large. Therefore, a large force would have to be applied for breaking the pile of tiles. 

When a tree is shaken, its fruits and leaves fall down

It is due to the inertia of rest in which an object remains in a state of rest or of uniform motion in a straight line unless compelled to change that state by an applied force.
When a tree is shaken, it brings a motion in the tree but its fruits and leaves tend to continue in the same state of rest because of its inertia which causes the leaves and fruits to separate from its branch and fall down. 

The action and reaction forces for a man walking on the road

According to Newton's third law of motion when one object exerts a force on another object, the second object instantaneously exerts a force back on the first. These two forces are always equal in magnitude but opposite in direction. when we push the road below backward, the road exerts an equal and opposite reaction force on our feet to make us move forward.

Derivation of the relation between force and acceleration

The mathematical formulation of the second law of motion:
If the mass of an object is 'm' and moving along a straight line with an initial velocity u. 

It is uniform velocity is v in time 't' and the constant force is F, 
The initial and final momentum of the object will be
 p₁ = mu and p₂ = mv respectively. 
The change in momentum = p₂ – p₁ = mv – mu 
 = m × (v – u). 
The rate of change of momentum = m (v - u )/ t 

According to Newtons second law
The applied force F ∝ m  (v−u ) /t 
F= km (v- u)/ t 
F = kma :  Here [a (acceleration)= (v – u)/t ] 
The quantity, k is a constant of proportionality. If the value of the constant, k becomes one. For this, one unit of force is defined as the amount that produces an acceleration of 1 m/s² in an object of 1 kg mass. 
That is, 1 unit of force = k × (1 kg) × (1 m/s²). 
Thus the value of k becomes 1. 
From Eq. F = ma 
The unit of force is kg m/s² or Newton N.

How a force can change the velocity of a body? 

According to newtons 2nd law, F = ma, therefore, a force can cause acceleration on a moving body and acceleration means a change in velocity at a constant rate i.e, a force can change the velocity of a body E.g., a moving ball comes to rest
The frictional force acting in the opposite direction of motion between the moving body and floor.

Newton’s third law of motion 

According to Newton's third law of motion when one object exerts a force on another object, the second object instantaneously exerts a force back on the first. These two forces are always equal in magnitude but opposite in direction.
If we are standing at rest and intend to start walking on a road. we must accelerate, and this requires a force in accordance with the second law of motion. We push the road below backward. The road exerts an equal and opposite reaction force on our feet to make us move forward.

Action and reaction forces 

Action and reaction forces act on different bodies. According to Newton's third law of motion when one object exerts a force on another object, the second object instantaneously exerts a force back on the first. These two forces are always equal in magnitude but opposite in direction. These forces act on different objects and never on the same object. These two opposing forces are also known as action and reaction forces.

The physical principle on which jet aircraft work

Newton's third law is the physical principle on which jet aircraft work. According to Newton's third law, when jet air crafts exert force on the air, air gives equal and opposite reaction force on the jet, due to which jet aircraft can move in the forward direction.

Recoil

When a gun is fired, it exerts a forward force on the bullet. The bullet exerts an equal and opposite reaction force on the gun. This results in the recoil of the gun. Since the gun has a much greater mass than the bullet, the acceleration of the gun is much less than the acceleration of the bullet.

Factors on which solid and liquid pressure depends

Factors of solid pressure: The pressure exerted on a surface depends on
(i) Force or thrust and - Larger the force on the surface, larger is the pressure exerted on it 
(ii) Area on which thrust is applied- Larger the area, less is the pressure exerted on it. 
Factors of liquid pressure : The pressure at a point inside the liquid depends on 
(i) depth of the point below the surface- Larger the depth of the point, the larger is the pressure exerted on it 
(ii) the density of the liquid- Larger the density of  the liquid, larger is the pressure exerted on it 
(iii) acceleration due to gravity- Larger the acceleration due to gravity larger is the pressure exerted on it.  

MOTION

Motion 

If the position of an object changes with time and its surroundings, the object is said to be in motion. The nature of the motion of an object can be studied by plotting of distance-time graph.

Displacement 

 It is the shortest distance between the initial and final positions traveled by the object. 

• It is not necessary that the distance covered by body is path length and it is always positive.
• If the motion is along a fixed direction, the distance and displacement of a moving object have the same magnitude. 
  
• When the object moves along the straight line in the same fixed direction.
• Displacement can not be greater than the distance traveled by an object, because displacement is always less than or equal to the distance traveled, if they are the same direction, then their magnitude will be equal, however, if they have different directions then displacement will be less than the distance.
• It is a vector quantity. 
• S.I unit is m. 

Speed 

The distance covered by a body in a unit interval of time is called speed. It is a scalar quantity. 

speed = Distance / Time

S.I unit is m/s

Uniform and nonuniform speed

If a body covers equal distance in equal interval of time throughout its motion is said to be uniform speed and if If a body covers the unequal distance in equal interval of time throughout its motion is said to be nonuniform speed.

Average speed

The ratio of the total distance traveled by the body to the total time of the journey is called average speed.
Av. Speed = Total distance / Total time.

Velocity 

The distance traveled by a body in a specified direction is called velocity. It is a vector quantity. S.I unit is m/s. 
Velocity = Displacement / Time.

Uniform velocity and non-uniform velocity

If a body covers equal distance in equal interval of time throughout its motion along a particular direction, is said to be uniform velocity and if If a body covers the unequal distance in equal interval of time throughout its motion in a particular direction is said to be non-uniform velocity. 

Average velocity 

The ratio of the total displacement traveled by the body to the total time of the journey is called average velocity.

Av. velocity = Total displacement / Total time.

Instantaneous velocity 

 When a body moving with a variable velocity, the velocity of the body at any instant is called instantaneous velocity.

Odometer

It measures distance traveled by automobile.

Speedometer - 

Rest and motion relative terms

Rest and motion are relative terms. While sitting on a moving bus our distance from the walls, roof, and floor of the bus does not change. So with respect to the bus our position does not change. Therefore we are at rest with respect to the bus but our distance from the bus stand changes with time. So we are moving with respect to the bus stand. So motion and rest are relative terms.

Differences between speed and velocity.

• Speed is the rate of change of motion and velocity is the rate of change of motion in a particular direction.
• Speed is scalar quantity and velocity is a vector quantity.
• Speed cannot be zero or negative but velocity can be zero, positive or negative.

Circular motion is accelerated motion

A particle moving in a circular path changes its direction continuously. its velocity therefore not constant even if its speed is constant. so it is an example of accelerated motion.

Expression of circular motion (v) = 2πr/t   Where r= radius and t = time.

Uniform circular motion.

If a body moves in a circular path and covers equal distance in equal interval of time, motion is called uniform circular motion.

Two examples are: Revolution of earth and the revolution of the moon.

Scalar and vector quantity.

• A scalar is a quantity that has only magnitude or numerical value. Ex- Temperature and mass are scalar quantity. 
• A vector is a quantity that has both magnitude and direction. Ex-Velocity and acceleration is a vector quantity.

Acceleration 

The rate at which the velocity of an object changes, is called the acceleration of the object, so when the particle moving with a uniform velocity its acceleration will be zero.

 Acceleration (a) = Change in velocity/time.

S.I unit is m/s^2.

Negative acceleration 

An object which moves in the positive direction has a positive velocity and if the speed of the object decreases, it is called negative acceleration or deceleration or retardation.

Uniform acceleration and nonuniform acceleration

When the equal change in velocity takes place in an equal interval of time, the acceleration is called uniform acceleration.

When an unequal change in velocity takes place in an equal interval of time, the acceleration is called non-uniform acceleration.

Retardation ( deceleration)

If the velocity of a body decreases with time, the motion is said to be declaration or retardation.

Differences

Distance and displacement.

• Distance is the length of the path traversed by the object in a certain time but displacement is the shortest distance between the initial and final position of the object.
• Distance is a scalar quantity, it has only the magnitude but displacement is a vector quantity and has both the magnitude and direction.
• Distance is always positive but displacement can be positive, negative or zero.
• The distance can be more than or equal to the displacement but displacement can never be greater than the distance, it can be equal or less than distance.

Speed and velocity
Difference between speed and velocity - 

• The rate of motion of an object or the distance traveled by the object in unit time is called speed. The SI unit of speed is meter per second but the velocity is the rate of motion of an object or the distance traveled by the object in a particular direction in unit time is called velocity.
• Speed is a scalar quantity but velocity is a vector quantity.
• Speed is always positive but velocity can be positive, negative or zero.

Uniform and non-uniform motion.
Uniform and non-uniform motion- If a moving object covers equal distances in equal intervals of time, it is said to be in uniform motion. but if the object covers unequal distances in equal intervals of time, it is said to be in non-uniform motion. 

Derive graphically the first equation of motion:

V = u + at
S= Ut + 1/2 at²
v² - u² = 2as.

1. DERIVATION OF EQUATION FOR VELOCITY - TIME RELATION :

Consider the velocity-time graph of an object that moves under uniform acceleration as shown in Fig. From this graph:
The velocity changes at a uniform rate. 
In Fig. the initial velocity (u) is represented by OA, the final velocity(v)is represented by BC and the time interval t is represented by OC.
BD = BC – CD (the change in velocity in time interval t)
AD parallel to OC and OC = AD = t 
From the graph, we observe that:
 BC = BD + DC
= BD + OA
Substituting BC = v and OA = u, we get 
v = BD + u or BD = v – u
The acceleration of the object is given by
a = Change in velocity /time taken 
= BD = BD = BC - DC
   AD     OC        OC
a = BC - DC
          OC
Substituting OC = t, BC = v and DC = u, we get 
a =  v - u
          t
or v - u = at
v = u + at 

2. DERIVATION OF EQUATION FOR POSITION-TIME RELATION 

Let us consider that the object has traveled a distance s in time t under uniform acceleration a. In Fig. 8.8, the distance traveled by the object is obtained by the area enclosed within OABC under the velocity-time graph AB. Thus, the distance s traveled by the object is given by 
s = area OABC (which is a trapezium) 
= area of the rectangle OADC + area of the triangle ABD
= OA × OC + 1/2 (AD × BD)
s = u × t + 1/2x t×at   (We have v = u + at and at = v-u =BD, therefore BD=at)
s = ut + 1/2 at²

3. DERIVATION OF EQUATION FOR POSITION–VELOCITY RELATION

From the velocity-time graph shown in fig,the distance s travelled by the object in time t, moving under uniform acceleration a is given by the area enclosed within the trapezium OABC under the graph. 
s = area of the trapezium OABC =1/2 (OA + BC) ×OC 
Substituting OA = u, BC = v and OC = t, we get
s = 1/2(u+v) t 
We have v = u + at 
at= v - u 
t = v - u
        a
s = 1/2 (v+u)(v-u)
                   a
2as = (v+u)(v-u)= v² - u²
2as = v² - u²

DERIVATION MATHEMATICALLY
V = u + at
S= Ut + 1/2 at²
v² - u² = 2as.

1. DERIVATION OF v = u + at

Let the time be t, Initial velocity u and final velocity v of a moving body.
We know that acceleration is the rate of change in motion.
a = final velocity - initial velocity / time =  v - u /t
at = v - u

v = u + at ..... equation (i)

2. DERIVATION OF S= Ut + 1/2 at²

We know that Average velocity = Total distance / Total time 

= Initial velocity + final velocity
                  2
= u + v 
      2
Distance s = Av. velocity x time = (u + v)/2 x t
2s = (v + u) x t
2s = (u + at + u) x t   [ v = u + at ]
2s = 2ut + at²
s = ut + 1/2at²..... equation (ii)

3. DERIVATION OF v² - u² = 2as.

We know that Average velocity = Total distance / Total time 
= Initial velocity + final velocity
                  2
= u + v 
      2
Distance s = Av. velocity x time = (u + v)/2 x t
2s = (v + u) x t
2s = (v+ u) x (v - u)/a      [ v = u + at and t = v - u/a]
2s = v² - u² /a
2as = v² - u² ..... equation (iii)