## Tuesday, October 03, 2017

Chap – 5
Laws of Motion

Laws of Motion:
Newton’s 1st Law: According to Newton’s first law every body continuous in its state of rest or uniform motion in a straight unless compelled by net external force to change that stage.
Inertia: The tendency of a body to remain in the state of rest or uniform motion is called inertia.
Types of Inertia:
1. Inertia of motion: The tendency of a body to remain in a state of motion is called inertia of motion. E.g. when a moving bus suddenly stops the passenger sitting in its feels a jerk formed due to inertia of motion.
2. Inertia of rest: The tendency of a body to remain in a state at rest is called inertia of rest. E.g. when a stationary bus suddenly starts moving then the passengers sitting in its feels a jerk in the backward direction due to inertial of rest.
3. Inertia of direction: The tendency of a body to keep on moving in the same direction motion is called inertia of direction. E.g. when a moving bus suddenly take a term then the passenger sitting in its experience a force acting away from the centre of the cured road linear.
Momentum:  The product of mass and velocity of a body is called momentum. It is denoted by i.e. Momentum is a vector quantity and its direction is same as that of velocity.

Or
The quantity of motion contained in a motion is known as linear momentum of a body. The S.I. unit ofis.
Newton’s 2nd Law: According to this law the time rate of change of linear momentum is directly proportional to the applied force and takes place in the direction of the force.
I.e.

Which is the mathematical statement of Newton’s second law?
Newton’s 3rd Law: To every action, there is an equal and opposite reaction.
Note: Action and Reaction reacts on different body so they do not cancel each other: E.g. when a bullet is fired from a gun the move in forward direction while the gun move backward, the forward motion of the bullet is action while the backward motion of the gun is the reaction.
Laws of conservation of momentum: Total initial momentum is always equal to the total final momentum for an isolated system.
Proof: By using Newton’s 2nd Law
Newton’s 2nd law is mathematically given byfor an isolated system
Proof: (ii) By using Newton’s 3rd Law:

Let,
Let two bodies of A and B of massesandmoving with initial velocityandalong the straight line collide. After collision, let their velocities change toandrespectively. Force exerted by A by B on B and A.
Force exerted by A an B.
Also
Using equation (i) and (ii) and equation (iii) we gets
Total initial momentum equal to final momentum
Hence momentum is conserved.

Impulse: A force which acts for a very short duration of time is called impulse. Mathematically impulseis the product of force and time during which the force acts.
i.e.
Units of Impulse is
Impulse momentum theorem: According to this theorem the impulse of a force is equal to the change in momentum of a body i.e. .

Proof: Newton’s second law can be mathematically written as
.
Q. Explain why
1. A cricket player lowers his hands while catching a ball.
2. Vehicles are fitted in shockers.

1. Ans. A cricket player lowers his hands while catching a ball→ when the ball is caught the impulse received by the hands is equal to the product of the force exerted by the ball and the time taken to complete the caught by moving the hands backward, the cricketers increases the time of the caught. The force exerted on the hand become much smaller and in does not hurt him.
2. Ans. When the vehicle moves on a rough road, it received a jerk. The shockers increases the time of jerk and hence reduces its force. This makes journey safe and comfortable of the vehicle from damage due to rough road.
Pulley Problem: Let two massesandbe connected to the two ends of the owing passing over a friction less pulley. Let T be the tension of the string. It, thenwill move down with acceleration a.

True Body diagram of
Free body diagram of

Recoiled velocity of a gun: The velocity with which a gun moves backward after firing a bullet is known as Recoiled Velocity.

Let a gun of massfire a bullet. Letbe the velocity of the bullet of mass. Letbe velocity of the gun with which it moves backward.
According to laws of conservation of linear momentum, we have
(Negative sign shows that the velocity of gun is in the opposite direction to that of the velocity of the bullet). Recoiled velocity of a gun is inversely proportional to the mass of the gun. Show-heavy gun recoils with small speed and light gun recoils with large speed).
Friction: The opposing force which comes into play whenever there is relative motion between any two surfaces in contact is called frictional force. In other words force acting on a object in a direction opposite to its motion relative to the surface.
Frictions are of two types:
1. Static friction
2. Kinetic friction.
The Static friction: The frictional force which acts between the body and the surface during the stationary state of the body is called static friction.
The Kinetic friction: The force of friction which acts between two surfaces of contact when there is a relating motion between them is called Kinetic friction.
Or
The force of friction which comes into play when one body moves over the surface of another body is called dynamic or kinetic friction.
Motion
Motion
Types of Kinetic friction:
1. Rolling friction: Force of friction which comes into play when one body rolls over the other body is called rolling friction. E.g. when a wheel roles over the road, rolling friction comes into play.
2. Sliding friction: The force of friction which comes into play when one body slides over the other body is called sliding friction. E.g. when a wooden black is pulled or pushed over the ground, then sliding friction comes into play.
Cause of friction:

The surface which looks smooth to the necked age has number of irregulaties over the surface which can be seen by a high powers microscope. So if we place a body over such a surface there will be very contact point between them. The pressure at that point is very high, so a kind of cold welding takes place. The force which is requires to breaks this cold welding points is called frictional force.
Limiting friction: The maximum value of static friction which comes into play when a body is just going to start sliding over the surface of another body is called limiting force of friction.
Or
The maximum value of static friction is called the limiting friction or limiting force of friction.
Laws of friction:
1. The maximum value of static friction is directly proportional to normal reaction i.e.

1. Force of friction is always opposite to the direction in which the motion tends to take place.
2. Frictional force depends upon the nature of the surface in contact.
Variation of force of friction with applied force: From the graph we find that initially the static friction increases proportionally with applied force. The friction force then reaches the maximum value which is called the limiting friction.

On increasing the applied force further, the body starts moving and the static condition is converted into Kinetic friction. Kinetic friction F max as seen in the graph is a constant friction.
Angle of friction:

(The angle made by resultant of limiting friction and normal reaction is called angle of friction). It is denoted by

Co=efficient of static friction may be defined as the tangent of the angle of friction.
Co-efficient of Static friction: According to the law of friction, the force of limiting friction is directly proportional to normal reaction.
i.e.
, whereis the constant of proportionally and it is called the co-efficient of static friction.
IfNewton then
Co-efficient of static friction may be defined as the force of limiting friction when the normal reaction is unity    (i.e. = 1).

Co-efficient of Kinetic friction: Force of kinetic friction is directly proportional to normal reaction
(i.e.
, whereis the constant of proportional and it is called the co-efficient of Kinetic friction.
IfNewton then
Co-efficient of kinetic friction may be defined as the force of kinetic friction when the normal reaction is unity.
Angle of Repose:

The minimum angle made by the inclined plane with the horizontal surface such that the body on the inclined plane is just ready to slide down along the inclined plane is called angle of repose.
From the above figure we have
Co-efficient of static friction may be defined as the tangent of the angle of repose.
We also have,
From equation (iii) and (iv)
Angle of Friction = Angle of Repose.
(Derive the expression for angle of Repose and hence prove that angle of repose is equal to angle of friction)

Acceleration of a body sliding down an inclined plane:

Let a body of massbe placed on a inclined plane with angle of inclination. In this case the body will slide down the plane with acceleration, a.
For such a situation,
Also we have
Which is the expression for acceleration of a body sliding down on inclined plane?
Q. Explain it is easier to pull than to push a body.

Let a forceis resolved into two components (i) , (ii) . The normal reaction, . Now, Kinetic force of friction is given by –
On the other hand, the same forceis applied to push a body of weightthan the normal reaction,
Kinetic force of friction is given by –
From equation (i) and (ii), it is clear that. That is force of friction in case of push is greater than that in case of pull. Hence, it is easier to pull them to push.
Method of reducing friction:
1. By polishing the surface.
2. Lubrication.
3. By providing the stream lined shape.
4. Converting sliding friction into rolling friction.
Circular motion:
Angular displacements: The angles describe the particle about the axis of rotation in a given time is called angular displacements i.e. it is denoted by.
The unit of angular displacement is radian.
i.e.

Angular speed: The ratio of angular displacement to the time taken is called angular speed. It is denoted by(omega).
I.e.
Unit is
Time Period: Time taken by a particle to complete one rotation is called time period. It is denoted by T.
Solution between W and T: Let a particle completes one rotation in time T. Then angular speed of the particle is given by –
Frequency: The no. of rotation completed by a particle in one second is called frequency. It is denoted by.
i.e.
Unit of frequency is per second or cycle per second or
Angular velocity in terms of frequency: We know
Angular velocity: It is a vector quantity whose magnitude is equal to the angle of speed that isorand direction is along the axis of rotation is an axial vector.

Relation between linear speed and angular speed:

Consider a particle moving in a circular path of radius r. Let it travels a distancein timewith linear speedand angular speed. Ifbe its angular displacement in time intervalthen
Linear speed of the particle is given by
But
Using equation (i) we get
In vector form equation (iii) can be written as
Uniform circular motion: (Centripetal acceleration) when a body moves in a circular path with a constant speed, then the motion of the body is known as uniform circular motion.
Other form of centripetal acceleration
We know
Angular acceleration: Angular acceleration is defined as the rate of change of angular velocity of a particle.
is angular velocity
Angular acceleration, or
Unit is rate per.

Relation between angular acceleration and linear acceleration
Linear acceleration of the particle is,
But
Equation (i) in vector form
When a particle rotates about an axis with an uniform angular velocity, then
Hence,
It means angular acceleration is zero when angular velocity of the particle is uniform or constant.

Comparison of equation of motion of linear and circular motion:
 Linear motion Circular motion (i) (ii) (iii) (iv) (i) (ii) (iii) (iv)

Centripetal force: The force which deviates an moving body from its linear path to move it along a circular path and is directed readily on work (i.e. towards the centre of circular path) is called centripetal force.
Expression for centripetal force:
If W be the angular velocity then
Now, whereis time period of the particle.
Since
Equation (i) to (iv) are different form of centripetal force.

Centrifugal force: A force that tends to move the body or particle of a rotating frame away from the axis of rotation is called centrifugal force. The magnitude of this force is equal to the magnitude of centrifugal force.
i.e.
Some examples of circular motion are as follows:
1. Car on a level circular road: Let us consider a car of mass m moving in a circular track or path of radius r with velocity v. The various forces acting on the car are:
1. Weight of the body, i.e. mg in the vertical downward direction.
2. Normal reaction force i.e. N in the vertical upward direction.
3. Functional force i.e. towards the centre of the circular track.
The car is moving on a circular path, so it requires centripetal force i.e. . The centripetal force must be provided by force of friction between types of the car and ground.
Can move in a circular track of radius r without sleeping
Banking of roads: Raking outer edge of the road higher than the inner edge at circular turns is called banking of roads.
1. Motion of a banked circular road: Let us consider a car moving on a banked circular road with an angle of banking Q. The various forces acting on the car are:
1. Weight of the car in the vertical downward direction.
2. Normal reaction perpendicular to the road directed upward.
3. Limiting force of frictionalong the road, towards the inner side. Resolvingand N in its components we final that, the componentsandare directed towards the centre of the circular track as therefore we can write that:
Which is the maximum velocity with which a car move in a banked circular road without Kiting
If fraction is not present, therefore

But for leveled circular road,
Which clearly means that in a absence of frictional force a car cannot take turn in a leveled circular road.