FORCE AND LAWS OF MOTION | Study

Effects of Force

• Force can change the shape and size of an object.
• Force can move a stationary object.
• Force can change the speed of a body.
• Force can stop a moving body. 
• Force can change the direction of a moving object.

Net or Resultant Force: 

Resultant Force or Net Force acts on a body if two or more forces act on it at the same time. Resultant Force or Net Force on a body is defined as the net effective force due to the multiple forces acting on it simultaneously. 

 Based on Net force, Forces are classified into two types as:

(A) Balanced forces

(B) Unbalanced forces

(A) Balanced Forces

• If the resultant of applied forces is equal to zero, the forces are called balanced forces.

• Balanced forces do not cause any change in state of an object. 

• Balanced forces can change the shape and size of an object. 

For example, when forces are applied from both sides over a balloon, the size and shape of the balloon is changed.

(B) Unbalanced Forces

• If the resultant of applied forces are greater than zero, the forces are called unbalanced forces. 

• Unbalanced forces can do the following :

* Move a stationary object

* Increase the speed of a moving object

* Decrease the speed of a moving object

* Stop a moving object

* Change the shape and size of an object 

Laws of Motion :

Galileo Galilei :

 Galileo Galilei was the first to say that objects move with a constant speed when no forces act on them. 

That is, if there is no unbalanced force acting on the object, the object moves forever with a constant speed without changing its direction.

In other words, if an object is moving on a frictionless path and no other force is acting upon it, the object moves forever with a constant speed without changing its direction.

Galileo’s Experiment:

Galileo’s thought experiment considered rolling balls on inclined planes in the absence of friction or other resistant forces. 

Galileo arranged two inclined planes opposite to each other as shown.

He rolls down the ball from the first inclined plane to climb the second inclined plane.

Galileo observations:

Galileo observed that:

• The ball rolling down the first inclined plane comes to rest after climbing a certain height on the second inclined plane.
• The speed acquired by the ball moving down a plane from a height is sufficient to enable it to reach the same height when climbing up another plane at a different inclination . 
• As the angle decreases, the body should travel a greater distance. 

From these observations, Galileo hypothesized as:

• if the force acting on the ball is only gravitational force, the height reached by the ball must be equal to the height from which it was rolled. 
• When the inclinations of the two planes are the same, the distance travelled by the sphere while rolling down is equal to the distance travelled by it while climbing up.
• Now, if the inclination of the second plane is decreased slowly, then the sphere needs to travel over longer distances to reach the same height. 
• If the second plane is made horizontal, then the sphere must travel forever trying to reach the required height.

This is the case when there is no unbalanced force acting on it. 

From his experiments Galileo proposed that the body could travel indefinitely far as , contrary to the Aristotelian notion of the natural tendency of an object to remain at rest unless acted upon by an external force. 

Therefore, Galileo can be credited with introducing the concept of inertia, later exploited by Newton.

However, in reality, frictional forces bring the sphere to rest after it travels over a finite distance.

After further study, Newton, in his first law of motion, stated that all objects resist a change in their natural state of motion.

This tendency of resisting any change in the natural state of motion is called “inertia”.

Newton’s Laws of Motion:

Newton studied the ideas of Galileo and gave the three laws of motion. These laws are popular as Newton’s laws of motion.

Newton’s First Law of Motion (Law of Inertia):

Any object remains in the state of rest or in the state of uniform motion along a straight line, until it is compelled to change its state by applying an external force.

Newton’s First Law of Motion in Everyday Life:

(a) A person standing in a bus falls backward when the bus starts suddenly. 

This happens because the person and bus both are at rest while the bus is not moving, but as the bus starts moving, an external force is acted by the bus on the legs of the person. This external force moves legs along with the bus. But the rest of his body has the tendency to remain in rest known as inertia of rest. Because of this, the person falls backward; if he is not alert.

(b) A person standing in a moving bus falls forward if the driver applies brakes suddenly. This happens because when the bus is moving, the person standing in it is also in motion along with the bus. But when the driver applies brakes the speed of the bus decreases suddenly or the bus comes to a state of rest suddenly, in this condition the legs of the person which are in contact with the bus come to rest while the rest of his body have the tendency to remain in motion. Because this person falls forward if he is not alert.

(c) Before hanging the wet clothes over the laundry line, usually many jerks are given to the clothes to get them dried quickly. Because of jerks, droplets of water from the pores of the cloth fall on the ground and the reduced amount of water 

in clothes dries them quickly. This happens because when suddenly clothes are made in motion by giving jerks, the water droplets in it have the tendency to remain in rest and they are separated from clothes and fall on the ground.

(d) When the pile of coins on the carrom-board is hit by a striker, the coin only at the bottom moves away leaving the rest of the pile of coins at the same place. This happens because when the pile is struck with a striker, the coin at

the bottom comes in motion while rest of the coin in the pile has the tendency to remain in the rest and they vertically falls the carrom-board

and remain at the same place.

Momentum

Momentum of an object at state of rest is zero :

Let an object with mass ‘m’ be at rest.

Since, object is at rest, its velocity, v = 0

We know that

Momentum, p is equal to the product of  mass, m and velocity, v = 0

⇒ p = m × 0 = 0

Thus, the momentum of an object in the rest i.e., non-moving, is equal to zero.

Unit of momentum :

SI unit of mass = kg

SI unit of velocity = meter per second i.e., m/s

We know that Momentum        (p) = m × v

    ⇒ p = kg × m/s

Or                                        ⇒    p = kg m/s

Therefore, SI unit of momentum = kg m/s

Impulse and Impulsive Force

If a cricketer catches a ball he moves his hand back while catching the ball. He does this to reduce the impact, due to the force of the ball on his hand. An object in motion has momentum. Momentum is defined as the product of mass and velocity of an object.

The momentum of the object at the starting of the time interval is called the initial momentum and the momentum of the object at the end of the time interval is called the final momentum. The rate of change of momentum of an object is directly proportional to the applied force.

Newton’s second law quantifies the force on an object. The magnitude of force is given by the equation,

F = ma, where ‘m’ is the mass of the object and ‘a’ is its acceleration. The CGS unit of force is dyne and the SI unit is newton (N).

A large amount of force acting on an object for a short interval of time is called impulse or impulsive force. Numerically impulse is the product of force and time. Impulse of an object is equal to the change in momentum of the object.

Impulse and Impulsive Force

The momentum of an object is the product of its mass and velocity. The force acting on a body causes a change in its momentum. In fact, according to Newton’s second law of motion, the rate of change in the momentum of a body is equal to the net external force acting on it.

Another useful quantity that we come across is “impulse”. “Impulse” is the product of the net external force acting on a body and the time for which the force is acted.

If a force “F” acts on a body for “t” seconds, then Impulse I = Ft.

In fact, this is also equal to the change in the momentum of the body. It means that due to the application of force, if the momentum of a body changes from “P” to “P ‘ ”, then impulse,I =  P ‘ – P.

For the same change in momentum, a small force can be made to act for a long period of time, or a large force can be made to act for a short period of time. A fielder in a cricket match uses the first method while catching the ball. He pulls his hand down along with the ball to decrease the impact of the ball on his hands.

In a cricket match, when a batsman hits a ball for a six, he applies a large force on the ball for a very short duration. Such large forces acting for a short time and producing a definite change in momentum are called “impulsive forces”.

Newton’s Second Law of Motion

Newton’s Second Law of Motion states that, the rate of change in momentum of an object is proportional to applied unbalanced force in the direction of force.

Mathematical expression:

State and derive newton’s second law of Motion

Statement:  Newton’s second law of motion states that the rate of change of momentum of an object is Proportional to the applied unbalanced force in the direction of force.

Derivation of Newton’s second law of motion:   

Suppose 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. 

⇒Initial momentum of the object, p₁ = mu 

⇒Final momentum, p₂ = mv 

⇒Change in momentum = p₂ – p₁

⇒The change in momentum = mv – mu

⇒The change in momentum = m × (v – u)

⇒The rate of change of momentum =  m(v -u)t

⇒  m (v -u)t

According to Newton’s Second Law of Motion,

Applied force α Rate of change in motion 

⇒ F   m (v -u)t

F=km (v -u)t = kma     —————————- (i)

Here, k is a constant of proportionality and 

(v -u)t is the rate of change of velocity, which equals acceleration, a.

The SI units of mass and acceleration are kg and m s-² respectively. 

The unit of force is so chosen that the value of the constant, K becomes one For this. 

One unit of force is defined as the amount that produces acceleration 

of 1 m s^-2 in an object of 1 kg mass. 

That is,

1 unit of force = k × (1 kg) × (1 m s^-2).

Thus, the value of k becomes 1. From Eq. (iii)

F = ma ————————————-

The unit of force is kg m s^-2 or newton, with the symbol N. 

Newton’s Third Law of Motion

To every action there is an equal and opposite reaction.

Applications:

(i) Walking is enabled by 3rd law.

(ii) A boat moves back when we deboard it.

(iii) A gun recoils.

4.  Rowing of a boat.

Law of Conservation of Momentum

Law of conservation of momentum states that, if two or more bodies collide, the sum of the initial momentum is equal to the sum of the final momentum.

Or

Law of conservation of momentum states that the sum (total) of the individual momentums of the colliding bodies just before the collision is equal to the sum (total) of the individual momentums of the colliding bodies after the collision

Derivation of Law of Conservation of Momentum From Newton’s Third Law of Motion.

Answer:

For a system of bodies ( two or more bodies ), the total vector sum of momenta of all the bodies due to the mutual action and reaction remain unchanged as long as no external force is acted on the system.

Consider two bodies A and B of the masses m₁, m₂ moving with the initial velocities u₁, u₂ respectively. 

For a system, let, these masses collide and their velocities after collision are v₁, v₂ respectively. 

If ‘A’ applies a F on B for a time, t;

‘B’ applies a force –F on A for time t [according to Newton’s third law of motion].

Then,

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Therefore, the sum of momentum before impact is equal to the sum of the momenta after the impact represents the law of conservation of momentum.