Laws of motion ( Mechanics )

Newton's law of motion

In 1686 Isaac Newton announces three laws of motion which are 

First law

It states everybody continues to remain in it's state of rest or uniform nation in straight line, unless it is acted upon by some external force to change that state 

This law is also known as law of inertia and has two parts - static and dynamic

• When a body is at rest it continues to remain at rest unless it is disturbed by some force. 

If a pen is kept in a pen stand, it will remain there forever unless and until it is removed from the stand .

• According to this law if a body is moving with uniform velocity, it will continue to move with the same velocity in the same straight line unless it is acted by some force. However practically it cannot be demonstrated as when a body is moving there is always some frictional force and air resistance acting on it which will oppose the motion of the body and it will stop moving after sometimes.

Inertia

It is the property of a body due to which it can not by itself, change its state of rest for uniform motion without a force.

Second law

It states that the rate of change of momentum is proportional to the applied force and takes place in the direction of the applied force .

This enables us to measure of force and established the the fundamental equation of dynamics.

Let 

m is equals to mass of a body

u is equals to initial velocity of the body

v is equals to final velocity of the body

f is equals to constant acceleration

t is equals to time required to change of velocity

P is equals to force required to change the velocity

Initial momentum of the body is mu

Final momentum of the body is mv

Change of momentum of the body is mv-mu

Rate of change of momentum is m(v-u)/t

Now , rate of change of momentum is mf

P is directly proportional to mf

P is equal to kmf

Where k is a constant of proportionality

If we define unit force is dead force which produces unit acceleration in a body of unit mass

P is equals to mf

Third law

Its States to every action there is an equal and opposite reaction.

When two bodies are in contact then action means the force which the one body exerts on another body and reaction is a force which II body exerts on the first.

According to third law of these two forces are equal and opposite.

Momentum

It is the quantity of motion possessed by a body moving in a straight line and is the product of mass of the body and its velocity.

m is equals to mass of body

V is equals to velocity

Momentum is equal to mass into velocity

M is equals to m×V

The unit of momentum in SI unit is kg metre per second

Law of conservation of momentum

Its States date when two or more bodies collide the total momentum before impact is equals the total momentum after impact.

Recoil of gun

It is an example of law of conservation of momentum.

When a bullet is fired from a loaded gun then according to Newton's third law of motion reaction of the bullet acts on the gun.

As a result it moves backward with some velocity and hence the person who fires a gun experience a jerk or impact.

This backward movement of the gun is gold recoil of gun.

M is equals to mass of

V is equals to velocity with which the gun recoils

m mass of the bullet

v velocity of the bullet after firing

Before firing both the gun and the bullet is at rest and the velocity is zero.

Momentum of the gun and bullet before firing is zero

Momentum of the gun and bullet after firing is MV+mv

According to law of conservation of momentum

Momentum before firing is equals to momentum after firing

0=MV+mv

-MV=mv

V=-mv/M

The negative sign shows that the opposite to the direction of firing of the bullet.

Impulse and impulse force

If a force f x on a body for a time t then the product of force and the time is called impulse of the force.

Impulse is equals F×t

Impulse is equal to change of the momentum

Impulsive forces  

forces which act on a body for a very short period of time and yet produce considerable change aaj of momentum on the body d r called impulsive forces.

Example

Collision of two bodies

Blow of hammer on piles

The unit of impulse in SI is kg metre per second

D'alembert's Principle

It states that similar to the equations of static equilibrium equation of dynamic equilibrium can also be established by introducing inertia force in the direction opposite to the acceleration in addition to the real forces.

Motion of a lift

Consider a lift or elevator moving with uniform acceleration carrying some mass

Let m is mass carried by the lift

a is uniform acceleration of the lift

w is weight carried by the lift is equal to mg

T is tension in the cable supporting the lift

When the lift is moving upward

the net upward force which causes the upward motion of the lift is 

T- w ..................... first

Also the net upward force is equals to 

m×a......................second 

From first and second

ma is equal to T-w

T = m( g+a)

When the lift is moving downward

the net downward force which causes the motion in the lift is

= w-T ................third 

Again net downward force is

= m × a ...........fourth

From third and fourth

ma = w- T

T= w - na

T = m ( g-a)

Concurrent forces

The forces acting at the same point of body are called concurrent forces.

Equilibrium of concurrent forces

A number of concurrent forces acting on a body are said to be in equilibrium if there vector sum is zero or if these forces can be completely represented by the sides of a closed polygon taken in the same order.

Lamis theorem

it states that if three forces acting on a particle keep it in equilibrium then each force is proportional to the sine of the angle between other two forces.

Free body diagram

a diagram for each body of the system showing all the forces exerted on the body by the remaining parts of the system is known as the free body diagram.

Friction

whenever a body moves all tense to move over the surface of another body of force comes into play which acts parallel to the surface of contact and deposit the relative motion this opposition force is known as the friction.

Static friction

the force of the friction which comes into play between two bodies before one body actually starts moving over the other body is known as static friction.

 Static friction is a self adjusting force.

Limiting friction

the maximum force of static friction which comes into play when a body just starts moving over the surface of another body is called limiting friction.

Kinetic friction

the force of friction which comes into play when a body is in a steady motion over the surface of of another body is known as kinetic or dynamic friction.

Kinetic friction or dynamic friction is less than the limiting friction.

Coefficient of limiting friction

It is the ratio of limiting friction to the normal friction.

Coefficient of kinetic friction

It is the ratio of kinetic friction to the normal friction.

Angle of friction

it is the angle which the resultant of the limiting friction and the normal reaction makes with the normal reaction.

Angle of repose

it is the minimum angle that an inclined plane makes with the horizontal when a body placed on it just begins to slide down.

Centripetal force

it is a force required to make a body move along a circular path with a uniform speed.

It always acts  along the radius and towards the centre of the circular path.

Centrifugal force

It is a fictitious force acting radially outward on a particle moving in a circle and  is equal in magnitude to the centripetal force.