** Chapter#3**

**DYNAMICS**

“It is the branch of Physics which deals with causes of motion and their effects”

**LAW OF MOTIONS**

Newton formulated three laws of motion in his book.

**NEWTON FIRST LAW OF MOTIONS**

Newton’s first law of motion is also known as the Law of Inertia.

**STATEMENT**

“Everybody continues its state of rest or uniform motion in a straight path until it is acted upon by an external, or unbalance force to change its state of rest or uniform motion”.

**EXPLANATION**

This law consists of a two parts

(a) When body is at rest

(b) When body is moving with uniform velocity

**When Body is At Rest**

Newton’s Law states that when a body is at rest, it continues its rest unless we apply a force on it. When we apply a force, it changes its state of rest and starts moving along a straight line.

**When Body is in Motion**

Newton’s Law states that when a body is moving, it moves in a straight line with uniform velocity, but when we apply an opposite force, it changes its state of motion and come to rest.

**Examples**

• A body riding a push-bike along a leveled road does not come to rest immediately when we apply a force, it changes its state of rest and starts moving along a straight line.

• If a bus suddenly starts moving, the passengers standing in the bus will fall in the backward direction. It is due to the reason that the lower part of the passengers which is in contract with the floor of the bus is carried forward by the motion of the bus, but the upper part of the body remains at rest due to inertia and so the passengers fall in backward direction.

**SECOND LAW OF MOTIONS**

** STATEMENT**

“When a force acts on an object it produces an acceleration which is directly proportion to the amount of the force and inversely proportional to the product of mass”

**EXPLANATION**

It is well known fact that if we push a body with greater force then its velocity increases and change of velocity takes place in the direction of the force. If we apply a certain force F on a mass m, then it moves with certain velocity in the direction of the force. If the force becomes twice then its velocity will also increase two times. In this way if we go on increasing the fore there will be increase in velocity, which will increase the acceleration.

**DERIVATION**

According to the Newton`s Second law of motion when a force acts on an object it produces an acceleration which is directly proportion to the amount of the force.

a ∝ F

and inversely proportional to the product of mass.

a ∝ 1/m

Combining all:

a ∝ F/m

a = K F/m

If the Value of K is 1

so,

a = F/m

or

F = ma

**1. FORCE**

Force is an agent which produces motion in a body but some time force may not be succeeded to produce motion in a body so we can say that the force is an agent which produces or tends to produce motion in a body.

We can further say that:

Force is an agent which stops or tends to stop the motion of a body. In simple word we can also say that force is an agent which changes or tends to change the state of an object.

**2. MASS**

The quantity of matter contained in a body is called mass.

FORMULA

F = ma

m = F/a

**UNIT**

The unit of mass is Kilogramme (kg)

**3. WEIGHT**

It is a force with which earth attracts towards its centre is called weight.

**FORMULA**

W = mg

**UNIT **

The unit of weight is Newton (N).

**THIRD LAW OF MOTION**

” To every action there is always an equal and opposite reaction”

**EXPLANATION**

According to Newton’s Law of Motion, we have:

F (action) = – F (reaction)

The negative (-) sign indicates that the two forces are parallel but in the opposite direction. If we consider one of the interacting objects as A and the other as B, then according to the third law of motion:

F (AB) = – F (BA)

F (AB) represents the force exerted on A and F (BA) is the force exerted on B.

**Examples **

• We walk on the ground, we push the ground backward and as a reaction the ground pushes us forward. Due to this reason we are able to move on the ground.

• If a book is placed on the table, it exerts some force on the table, which is equal to the weight of the book. The table as a reaction pushes the book upward. This is the reason that the book is stationary on the table and it does not fall down.

**INERTIA**

**Definition**

“Inertia is the tendency of a body to resist a change in its state.”

**Examples**

Cover a glass with a post card and place a coin on it. Now strike the post card swiftly with the nail of your finger. If the stroke has been made correctly, the postcard will be thrown away and the coin will drop in the glass.

If a moving bus stops suddenly, the passenger standing in it feels a jerk in the forward direction. As a result he may fall. It is due to the fact that the lower part of the standing passengers comes to rest as the bus stops. But the upper portion remains in motion due to inertia.

DIFFERENCE BETWEEN MASS AND WEIGHT

**Mass**

1. The quantity of matter present in a body is called mass.

2. The mass of a body remains constant everywhere and does not change by change in altitude.

3. Mass of a body possesses no direction. So it is a scalar quantity.

4. Mass can be determined by a physical balance.

5. Unit of mass in S.I system is KILOGRAM (kg)

6. Mass of a moving body is m=F/a.

**Weight**

1. The force with which the earth attracts a body towards its centre is called the weight of the body.

2. The weight of a body is not constant. It is changed by altitude.

3. Weight of a body has a direction towards the centre of the earth. So it is a vector quantity.

4. Weight can be determined by only a spring balance.

5. Unit of weight in S.I system is NEWTON (N)

- Weight of a body is W = mg.

**MOMENTUM**

“The quantity or quality of motion is called momentum and it is denoted by P”

**MATHEMATICAL DEFINITION**

“It is the product of mass and velocity.”

**MATHEMATICAL REPRESENTATION**

P = mV

where:

• p is the momentum

• m is the mass

• v the velocity

**Unit**

Unit of momentum in S.I system is kgms^{-1}

**LAW OF CONSERVATION OF MOMENTUM**

The law of conservation of momentum is a fundamental law of nature, and it states that

“The total momentum of an isolated system of objects constant”.

One of the consequences of this is that the centre of mass of any system of objects will always continue with the same velocity unless acted on by a force outside the system

**EXAMPLE**

Consider two bodies A and B of mass m_{1} and m_{2} moving in the same direction with velocity U_{1} and U_{2} respectively such that U_{1} is greater than U_{2}. Suppose the ball acquire velocity V_{1} and V_{2} respectively after collision

Momentum of the system before collision = m_{1}U_{1} + m_{2}U_{2 }

Momentum of the system after collision = m_{1}V_{1} + m_{2}V_{2 }

According to the law of conservation of momentum:

Total momentum of the system before collision = Total momentum of the system after collision =

m_{1}U_{1} + m_{2}U_{2} = m_{1}V_{1} + m_{2}V_{2 }

**FRICTION**

Definition

“When a body moves over the surface of another body then the opposing force is produce and this opposing force is called force of friction”

**Explanation**

Suppose a wooden block is placed on a table and a spring balance is attached on it. If we apply a very small force of magnitude F by pulling the spring gradually and increase it, we observe that the block does not move until the applied force has reached a critical value. If F is less then critical value, the block does not move. According to Newton’s Third Law of motion an opposite force balance the force. This opposing force is known as the force of friction or friction.

**Causes of Friction**

If we see the surface of material bodies through microscope, we observe that they are not smooth. Even the most polished surfaces are uneven. When one surface is placed over another, the elevations of one get interlocked with the depression of the other. Thus they oppose relative motion. The opposition is known as friction.

**Factors on which Friction Depends**

The force of friction depends upon the following factors:

**1. Normal Reaction (R)**

Force of friction is directly proportional to normal reaction (R), which act upon the body in upward direction against the weight of the body sliding on the surface.

**2. Nature of Surfaces**

Force of friction also depends upon the nature of the two surfaces. It is denoted as u and has constant values for every surface. It is different for the two surfaces in contact.

**COEFFICIENT OF FRICTION**

The coefficient of friction is a number which represents the friction between two surfaces. Between two equal surfaces, the coefficient of friction will be the same. The symbol usually used for the coefficient of friction is µ.

The maximum frictional force (when a body is sliding or is in limiting equilibrium) is equal to the product of coefficient of friction and the normal reaction force.

F = µR

Where m is the coefficient of friction and R is the normal reaction force.

This frictional force, F, will act parallel to the surfaces in contact and in a direction to oppose the motion.

**Mathematical**

The value of limiting friction increases proportionally with the increase in normal reaction. Hence, liming friction F(s) is directly proportional to the normal reaction.

F(s) ∝ R

=> Fs =µR……….. (i)

µ = F(s)/R

µ is the constant of proportionality, which depends upon the nature of the surfaces of the two surfaces in contact. It is known as the coefficient of friction. It is only a number without any unit. We know that the normal reaction is directly proportional to the weight of the block, therefore,

R = W = mg (Substituting the value of R in equation (i))

=> Fs = µmg

**ADVANTAGES OF FRICTION**

1, We could not walk without the friction between our shoes and the ground. As we try to step forward, we push your foot backward. Friction holds our shoe to the ground, allowing you to walk.

2, Writing with a pencil requires friction. We could not hold a pencil in our hand without friction.

3, A nail stays in wood due to friction

4, Nut and bold can hold due to friction

**DISADVANTAGES OF FRICTION**

1, In any type of vehicle–such as a car, boat or airplane–excess friction means that extra fuel must be used to power the vehicle. In other words, fuel or energy is being wasted because of the friction.

2, The Law of Conservation of Energy states that the amount of energy remains constant. Thus, the energy that is “lost” to friction in trying to move an object is really turned to heat energy. The friction of parts rubbing together creates heat.

3, Due to the friction a machine has less frequency than 100%

4, Due to friction machine catch fire.

**Methods of Reducing Friction**

Friction can be reduced by the following methods:

1. The various parts of the machines that are moving over one another are properly lubricated.

2. In machines, the sliding of various parts is usually replaced by rolling. This is done by using ball bearings.

3. Where sliding is unavoidable, a thick layer of greasing material is used between the sliding surfaces.

4. The front of the fast moving objects, e.g. cars, aeroplanes are made oblong to decrease air friction.

**Rolling Friction**

If we set a heavy spherical ball rolling, it experiences an opposing force called rolling friction. When a body rolls over a surface, the force of friction is called rolling friction. Rolling friction is much less than the sliding friction. This is because the surfaces in contact are very much less.

**Centripetal Force **

Definition

“The force that causes an object to move along a curve (or a curved path) is called centripetal force.”

**Mathematical Expression**

We know that the magnitude of centripetal acceleration of a body in a uniform circular motions is directly proportional to the square of velocity and inversely proportional to the radius of the path Therefore,

a_{c} ∝ v^{2}

a_{c} ∝ 1/r

Combining both the equations:

a_{c} ∝ v^{2}/r (From Newton’s Second Law of Motion)

F = ma

=> F_{c} = mv^{2}/r

Where,

• F_{c} = Centripetal Force

• m = Mass of object

• v = Velocity of object

• r = Radius of the curved path

Factors on which Fc Depends:

Fc depends upon the following factors:

• Increase in the mass increases F_{c}.

• It increases with the square of velocity.

• It decreases with the increase in radius of the curved path.

**Examples**

• The centripetal force required by natural planets to move constantly round a circle is provided by the gravitational force of the sun.

• If a stone tied to a string is whirled in a circle, the required centripetal force is supplied to it by our hand. As a reaction the stone exerts an equal force which is felt by our hand.

• The pilot while turning his aeroplane tilts one wing in the upward direction so that the air pressure may provide the required suitable F_{c}.

**Centrifugal Force**

** Definition**

“A force supposed to act radially outward on a body moving in a curve is known as centrifugal force. “ OR

“The force that draws a rotating body away from the center of rotation”.

**Explanation**

Centrifugal force is actually a reaction to the centripetal force. It is a well-known fact that Fc is directed towards the centre of the circle, so the centrifugal force, which is a force of reaction, is directed away from the centre of the circle or the curved path.

According to Newton’s third law of motion action and reaction do not act on the same body, so the centrifugal force does not act on the body moving round a circle, but it acts on the body that provides F_{c}.

**Examples**

• If a stone is tied to one end of a string and it is moved round a circle, then the force exerted on the string on outward direction is called centrifugal force.

• The aeroplane moving in a circle exerts force in a direction opposite to the pressure of air.

• When a train rounds a curve, the centrifugal force is also exerted on the track.