Newtons Law of Motion and Frictions

Cause of Motion

When a body is initially at rest or start to move, a question that arises in our mind is ‘what caused the body to move? Similarly, when a moving object comes to rest, you would like to know what brought it to the rest? If a moving object speeds up or slows down or changes its direction. What speeds up or slows down the body? What changes the direction of motion? 

Example: 

A stone released from the top accelerates downward due to the gravitational pull of the earth. 

A force is required to put a stationary body in motion or stop a moving body and some external agency is needed to provide this force. The external agency may or may not be in contact with the body. 

1.0Force

A force can be defined as ‘a push or a pull exerted on an object that can cause the object to speed up, slow down, change the direction as it moves and it can change its shape and size’.

An interaction of one object with another object results in a force between the two objects i.e., to apply force at least two objects are required.

The effect of a force depends on both magnitude and direction; thus, force is a vector quantity. A force vector points in the direction of the force, and its length is proportional to the magnitude of the force.

Forces applied on an object in the same direction add to one another. If two forces act in the opposite directions on an object, the net force acting on it is the difference between the two forces. 

Types of Forces

Contact force :  It is a force that is exerted only when two objects are touching each other.  

Examples :

Muscular Force : The force resulting due to the action of muscles is known as the muscular force.

Friction : Friction is a force that resists relative motion. Friction is found everywhere due to every material i.e., solids, liquids and gases.

Tension : Tension is a force exerted by string, ropes, fibres, and cables when they are pulled.

Normal force : The force perpendicular to the surfaces of the objects in contact is called normal force. 

Non-contact force : It is a force that one object exerts on another when they are not touching each other. 

Examples : 

Magnetic force : The force exerted by a magnet on a piece of iron or on an another magnet is called magnetic force. Like (or similar) poles repel while unlike (or opposite) poles attract each other.

Electrostatic force : The force exerted by a charged body on another charged body or uncharged body is known as electrostatic force. Like charges repel and unlike charges attract each other.

Gravitational force : The attractive force between two objects that have mass is called gravitational force. Force of gravity is always attractive in nature and pulls objects toward each other. A gravitational attraction exists between you and every object in the universe that has mass.

Balanced and Unbalanced Forces 

Balanced forces

If the resultant of all forces acting on a body is zero, the forces are called ‘balanced forces’. 

If the net force exerted on an object is zero, then the forces acting on it are said to be balanced. In such a case, the acceleration of the object is zero and its velocity remains constant. That is, if the net force acting on the object is zero, the object either remains at rest or continues to move with constant velocity.

Unbalanced Forces

If the resultant of all forces acting on a body is not zero, the forces are called ‘unbalanced forces’.

If the net force exerted on an object is not zero, then the forces acting on it are said to be unbalanced. In such a case, the acceleration of the object is not zero and its velocity changes. That is, if the net force acting on the object is not zero, then such a force changes state of rest or the state of uniform motion of the object.

An object acted upon by an unbalanced force changes speed and direction

Effects of Resultant force

A non-zero resultant force may produce following effects on a body :

(i) It may change the speed of the body.

(ii) It may change the direction of motion. 

(iii) It may change both the speed and direction of motion.

(iv) It may change the size and the shape of the body.

2.0Galileo's Inclined Planes 

Galileo studied motion of objects on an inclined plane. He noted that balls rolling down [see fig.(a)] the inclined planes picked up speed (i.e. acceleration), while balls rolling on [see fig.(b)] up the inclined planes loose speed (i.e. retardation). From this he reasoned that balls rolling on [see fig.(c)] a horizontal plane would neither speed up nor slow down. The ball would finally come to rest not because of its ‘nature’ but because of friction. 

Another experiment by Galileo leading to the same conclusion involves an ideal frictionless double inclined plane. A ball released from rest on one of the planes rolls down and climbs up the other. If the planes are smooth, the final height of the ball is nearly the same as the initial height (a little less but never greater). 

In the ideal situation, when friction is absent, the final height of the ball is the same as its initial height . If the slope of the second plane is decreased and the experiment is repeated, the ball will still reach the same height, but in doing so, it will travel a longer distance. Thus, the ball must roll a greater distance as the angle of the second inclined plane on the right is reduced. He argued that when the slope of the second plane is made zero i.e. it becomes a horizontal plane, the ball must travel an infinite distance since it can never reach its initial height on the first plane. In other words, its motion never ceases. This is, of course, an idealized situation. 

Galileo arrived at a new insight that ‘the state of rest and the state of uniform motion (motion with constant velocity) are equivalent’. In both cases, there is no net force acting on the body. A body does not change its state of rest or uniform motion, unless an external force compels it to change that state. The tendency of things to resist changes in motion was what Galileo called inertia.

Newton built on Galileo’s ideas and laid the foundation of mechanics in terms of three laws of motion that go by his name. Galileo’s law of inertia was his starting point which he formulated as the first law of motion.

3.0Newton's First Law of Motion (or Galileo's law of Inertia)

‘Every object continues in its state of rest, or of uniform motion in a straight line, unless it is compelled to change that state by forces impressed upon it’.

This law defines the force and states that the force is a factor which can change the state of object.

Definition of force from Newton's first law of motion "Force is the push or pull which changes or tends to change the state of rest or of uniform motion". 

Concept of Inertia

Inertia is ‘the natural tendency of an object to remain at rest or in motion at a constant speed along a straight line’. In other words, ‘the tendency of an object to resist any attempt to change its velocity’ is called inertia.

The mass of an object is a quantitative measure of inertia. More the mass, more will be the inertia of an object and vice-versa.

$\text{Inertia}\propto \text{mass}$

Inertia of an object can be of three types :

(1) Inertia of rest  

(2) Inertia of motion    

(3) Inertia of direction

Inertia of Rest

It is the tendency of an object to remain at rest. This means an object at rest remains at rest until a sufficiently large external force is applied on it. 

Examples of inertia of rest 

(1) When you are sitting in a stationary car, if the car starts suddenly i.e. accelerates forward, you feel as if your body is being pushed back against the seat, because your body which was initially at rest resists this change due to inertia. The lower part of the body comes in motion as it is in direct contact with the car floor, while the upper portion still remains at rest due to inertia of rest. If the speed of car increases slowly, you will not feel a push or a jerk because the inertia of motion will get transferred to the whole body.

(2) When a blanket is given a sudden jerk, the dust particles fall off. This is because the blanket suddenly comes in motion but the dust particles due to inertia of rest, still remain at rest. As a result, the dust particles get separated from the blanket.

(3) Set a five-rupee coin on a stiff smooth card covering an empty glass tumbler standing on a table. Give the card a sharp horizontal flick with a finger.

If we do it fast then the card moves away, allowing the coin to fall vertically into the glass tumbler due to its inertia of rest. 

(4) Make a pile of similar carrom coins on a table. Attempt a strong horizontal hit at the bottom of the pile using another carrom coin or the striker. If the hit is strong enough, the bottom coin moves out quickly. Once the lowest coin is removed, the inertia of the other coins makes them ‘fall’ vertically on the table.

This is because the lowest coin comes in motion while the other coins remain at rest due to inertia of the rest of the above coins. 

Inertia of Motion

It is the tendency of an object to remain in the state of uniform motion. This means an object in uniform motion continues to move uniformly until an external force is applied on it. 

Examples of inertia of motion 

(1) When you are driving a car and you apply brakes to stop the car suddenly, you feel as if your body is being pushed forward, because your body resists the decrease in speed. The lower part of the body comes to rest as it is in direct contact with the car floor, while the upper portion still remains in motion due to inertia of motion. If you stop the car by decreasing its speed slowly, you will not feel a push or a jerk because the inertia of rest will get transferred to the whole body.

(2) A person jumping out of a moving train has the tendency to fall forward. This is because of jumping, his feet come to rest as they touch the ground. But, his upper body continues to move forward due to inertia of motion.

(3) An athlete runs for some distance quickly before taking a long jump. As a result, he takes a longer jump due to inertia of motion.

(4) When you move a hammer with a loose hammerhead in downward motion and suddenly stop it on a floor or a wooden base, the hammerhead gets tightened. This is because the handle of the hammer suddenly comes to rest on hitting the floor, while the hammerhead continues to move downward due to more inertia of motion, and hence gets tightened. If you move the hammer slowly, the state of rest will get transferred to the hammerhead also, thus, the hammerhead will not get tightened. 

(5) A ball is thrown in the vertical upward direction by a passenger sitting inside a moving train. 

The ball will return :-  

Case I : In the hands of the passenger, if the train is moving with constant velocity.

Case II : Ahead the passenger, if the train is retarding means slowing down.

Case III : Behind the passenger, if the train is accelerated means speeding up.

Inertia of Direction

It is the tendency of an object to maintain its direction. This means an object moving in a particular direction continues to move in that direction until an external force is applied to change it.

Examples of inertia of direction

(1) When your motorcar makes a sharp turn at a high speed, you tend to get thrown to one side. You tend to continue in straight-line motion due to inertia of direction. 

(2) When a wheel rotates at high speeds, the sand particles on the wheel fly tangentially along a straight line due to inertia of direction.

4.0Momentum

The total quantity of motion possessed by a moving body is known as the momentum of the body. It is the product of the mass and velocity of a body. 

SI Unit : kg m s^–1 or Newton-second or N-s

C.G.S. unit : g cm/s or g cm s^–1

Dimension: [M L T^–1]

It is a vector quantity.

Linear momentum can be positive or negative depending on its direction.

● For a given velocity, the momentum is directly proportional to the mass of the object . This means more the mass, more will be the momentum and vice-versa. If a car and a truck has same velocity, then, the momentum of truck is more than the momentum of car as the mass of a truck is greater than the mass of a car .

● For a given mass, the momentum is directly proportional to the velocity of the object . This means more the velocity, more will be the momentum and vice-versa. If two bodies with same masses move with different velocities then, the body having more velocity will have more momentum [see fig (b)].

● For a given momentum, the velocity is inversely proportional to the mass of the object . This means smaller the mass, more will be the velocity of an object and vice-versa. If a car and a truck has same momentum, the velocity of car will be more than the velocity of truck as the mass of a car is smaller than the mass of a truck .

5.0Newton's Second Law of Motion

According to Newton’s second law, 

‘The rate of change of momentum of any system is directly proportional to the applied external force and this change in momentum takes place in the direction of applied force’.

Mathematical formulation of Newton’s second law of motion

Let an object of mass ‘m’ is moving along a straight line with an initial velocity ‘u’. It is uniformly accelerated to velocity ‘v’ in time ‘t’ by the application of a constant force F throughout the time t. The initial and final momentum of the object will be, 

p₁ = mu and p₂ = mv, respectively.

The change in momentum, Δp = p₂ – p₁ = mv – mu = m (v – u)

The force F is proportional to the rate of change of momentum, that is,

$F\propto \frac{\Delta p}{\Delta t}\text{or}F\propto \frac{m(v-u)}{\Delta t}$

$F=k\frac{m(v-u)}{\Delta t}$ ...(1)    

where, k is a constant of proportionality.

as, acceleration, 

$a=\frac{(v-u)}{\Delta t}$ ...(2)

From eq.(1) & eq.(2), we get, F = k m a   ...(3)

The SI units of mass and acceleration are kg and ms^–2 respectively. The unit of force is so chosen that the value of the constant ‘k’ becomes one. 

That is, 1 unit of force = k × (1 kg) × (1 ms^–2)

or  k = 1

Thus, the value of k becomes 1. Therefore, the eq.(3) reduces to,

F= ma

Units for measurement of force :

Newton's first law of motion defines force and second law of motion measures force. It gives the units, dimensions and magnitude of the force.

Unit of force = (Unit of mass) × (unit of acceleration) = 1kg × 1 m/s² = 1 Newton

1 Newton = 1 kg-m/s²

1 dyne = 1 g-cm/s²

Absolute units

(i) Newton (M.K.S)

(ii) dyne (C.G.S)

Gravitational / Practical units

kg - wt or kg - f (kg. force)

g - wt or g - f (gram force)

Relation between above units

1 kg-wt = 9.8 Newton

1 g-wt = 980 dyne

1 N = 10⁵ dyne

Dimensions of force  

[F]= [m] [a] = [M¹] [L¹T^–2] = M¹L¹T^–2

Applications of Newton’s Second Law

A cricketer catching the ball

(i) Suppose a light-weight vehicle (say a small car) and a heavy-weight vehicle (say a loaded truck) are parked on a horizontal road. A much greater force is needed to push the truck than the car to bring them to the same speed in same time. This is because, for a given time interval, the force is directly proportional to the change in momentum. Here, the change in momentum of truck is larger than that of the car, therefore, force required for a truck is larger as compared to that required for a car.

(ii) A cricketer lowers his hand while catching a ball. So that the time interval of momentum change is increased due to which the average reaction force on his hands decreases. So he can save himself from getting hurt.

(iii) If two stones, one light and the other heavy, are dropped from the top of a building, a person on the ground will find it easier to catch the light stone than the heavy stone. This is because the force is directly proportional to the mass of an object.

(iv) Speed/velocity is another important parameter to consider. A bullet fired by a gun can easily pierce human tissue before it stops, resulting in casualty. The same bullet fired with moderate speed will not cause much damage. Thus for a given mass, the greater the speed, the greater is the opposing force needed to stop the body. 

(v) When an athlete goes for a high jump, he is made to fall on a cushioned bed. This increases the time of falling of the athlete, thereby reducing the force exerted on him, causing no injury.

(vi) Shockers are provided in vehicles to avoid jerks.

(vii) Buffers are provided in bogies of train to avoid jerks.

● Impulse Momentum Theorem : The impulse of force is equal to change in momentum of body. This relation is known as impulse momentum theorem.

Units of impulse = N – s or kg-m/s.

Dimension of impulse = [F] [t] = [m] [a] [t] = [M¹L¹T^–2T¹] = [M¹L¹T^–1]

● When an object is moving along a circular path, its velocity is tangential to the circular path, hence, its momentum is also tangential to the circular path.

● The greater the change in the momentum in a given time interval, the greater is the force that needs to be applied.

● If force is constant i.e., F = ma = constant, then, the acceleration produced in the body, _{a\propto \frac{1}{m}}. That is, for a given force, acceleration produced is inversely proportional to its mass. 

● If same force F is applied to masses m₁ and m₂ and the resulting accelerations in them are a₁ and a₂ respectively, then,

m₁a₁ = m₂a₂ or  $\frac{a_2}{a_1}=\frac{m_1}{m_2}$

6.0Rocket Propulsion

In a rocket engine, the highly combustible fuel burns at a tremendous rate. The rocket exerts a downward (or backward) force on the exhaust gas and thus, according to Newton’s third law, the exhaust gas exerts an upward (or forward) force on the rocket, these forces are equal in magnitude. It is the reaction force of the exhaust gas that accelerates the rocket forward. That is why a rocket can accelerate even in outer space.

Case-I : If rocket is accelerating upwards, then -

Net upwards force on rocket = ma

$\frac{vdm}{dt}-mg=ma$

Case-II : If rocket is moving with constant velocity, then a = 0 

$\therefore v\frac{dm}{dt}-mg=m\times 0$

$v\frac{dm}{dt}=mg$

Solved Examples

1. Momentum of an object is 20 kg m s^–1. What will be its momentum if

(a) Its mass is doubled but the velocity remains the same?

(b) The velocity is reduced to (1/3) of its original magnitude but mass remains the same?

Solution

Initial value of momentum, p = mv = 20 kg m s^–1

(a) Final momentum, p' = m'v = (2m)(v) = 2mv = 2 × 20 = 40 kg m s^–1

(b) Final momentum, p'' = mv' = m(v/3) = (mv)/3 = 20/3 = 6.67 kg m s^–1

2. A 65 kg girl is driving a 535 kg car at a constant velocity of 11.5 m/s.  Calculate the momentum of the girl-car system.

Solution

The total mass of the system, m = mass of girl + mass of car = 65 + 535 = 600 kg

Now, momentum, p = mv = 600 × 11.5 = 6900 kg m s^–1

3. A cricket ball of mass 160 g is moving with a velocity of 30 m/s and is hit by a bat so that the ball is turned back with a velocity of 35 m/s.  If the duration of contact between the ball and bat is 0.01 s, find the impulse and the average force exerted on the ball by the bat.

Solution

According to given problem change in momentum of the ball 

Δp = p_f – p_i = m(–v – u) = 160 × 10^–3 [–35 – (30)]

So by impulse-momentum theorem, Impulse J = Δp = –10.4 N-s,

and by time averaged definition of force in case of impulse

$F_{\text{av}}=\frac{J}{\Delta t}=\frac{|\Delta p|}{\Delta t}=\frac{10.4}{0.01}=1040\text{N}$

4. A 750 kg rocket is set for a vertical firing. If the exhaust speed is 1200 m/s. Then calculate the mass of gas ejected per second to supply the thrust needed to overcome the weight of the rocket.

Solution

Force required to overcome the weight of rocket F = mg & thrust needed 

$=v\frac{dm}{dt}$

So, v $\frac{dm}{dt}$= mg

⇒ $\frac{dm}{dt}=\frac{mg}{v}=\frac{750\times 9.8}{1200}=6.125\text{kg/s}$