Linear Momentum, Physics tutorial

Momentum:

Define: The momentum of a body is stated as the product of its mass and its velocity.

When a body of mass m moves by a velocity 'v', then its momentum 'p' is given by:

P = mv

The SI unit of momentum is kilogram metre per second (kgms-1).

The Momentum is a vector quantity; its magnitude is the numerical value of the product m x v and its direction is in the direction of v.

Bodies moving in the straight line have 'linear momentum' as rotating bodies contain 'angular momentum'.

Impulse:

Define: Impulse is stated as the product of the average force on the particle and the time throughout which it acts.

When a force F acts for a short period of time t, the impulse I is given by:

I = F x t

Impulse is a vector having magnitude equivalent to the product F x t, and direction similar as that of the force F.

The unit of impulse is Newton-second (Ns).

Newton's Laws of Motion:

Newton's First Law of Motion:

Newton's first law of motion defines or illustrates that each and every object continues in its state of rest or of uniform motion in a straight line unless acted on by an external force.

Significance of Newton's first law:

1) The law recognizes inertia as an intrinsic property of a body at rest or moving by a constant velocity. Inertia is therefore a property of matter and might be stated as the reluctance of a body to modify its state of rest or of uniform motion in a straight line except an external force acts on it. The mass of a body is the measure of its inertia that is, the greater the mass of a body, the more its inertia and vice-versa.

2) Newton's first law describes why passengers in a fast moving vehicle tend to move forward if the vehicle stops all of a sudden or jerks backwards whenever the vehicle all of a sudden speeds off, as there is little or no force to restrain them. Due to this reason motorists are recommended to employ seat belts (or safety belts) while traveling in their cars.

3) Newton's first law as well describes what force does, however it does not advise how it must be measured.

Newton's Second Law of Motion:

The Newton's second law defines that the rate of change of momentum is directly proportional to the force causing motion and occurs in the direction of that force.

Significance of Newton's Second Law:

1) It lets us to state an absolute unit of force that remains constant under all conditions.

2) It offers a measure of force as product of mass and acceleration.

3) It provides an operational definition of force as the rate of change of momentum.

According to Newton's Second Law,

F = change in momentum/time taken for the change

F = (m)(Δv/Δt)

F = ma

In the S.I. unit system, 'F' is in Newton, 'm' in kilogram and the acceleration, 'a' in meter per sec squared (ms-2).

Newton's Third Law:

Newton's third law of motion defines or illustrates that to any action there is an equivalent and opposite reaction.

Illustration of Newton's third Law in operation:

Whenever you hit your head against the wall, you apply some force on the wall and the wall in turn applies an equivalent and opposite force on your head. If the force you applied on the wall is large, then pain you would feel on your head is a proof of Newton's third law. This means that forces are for all time paired.

No single force exists in isolation.

Conservation of Linear Momentum:

The main principle of conservation of linear momentum might be defined in various manners as follows:

1) In any system of colliding objects, the net momentum is for all time conserved given that there is no total external force acting on the system. 

2) The net momentum of an isolated or closed system of colliding bodies remains constant.

3) When two or more bodies collide in a closed system, the total or net momentum after the collision is equivalent to the total momentum before the collision.

The conservation of linear momentum (p = mv) is a significant concept in physics. In a closed system, if momentum is conserved then the initial and final total momentums are equivalent. By a 'closed system' we signify that there are no outside forces acting on the system. Let suppose that there are no significant outside forces--like friction--present throughout the collisions. This will not exactly be the case however when the collision is short in duration, it is reasonable to make this supposition as the friction is small and the period over which it acts is short.

Thus, ∑ pi = ∑ pf

Initial: p1 = m1u1 + m2u2

Final: p2 = m1v1 + m2v2

And lastly, conservation of momentum ∑ pi = ∑ pf gives

m1u1 + m2u2 = m1v1 + m2v2

Elastic and Inelastic Collision:

Elastic collision: This is the kind of collision in which both the momentum and kinetic energy of the system are conserved, and is termed as elastic collision. The collision among subatomic particles is usually elastic. The collision among the two steel or glass balls is almost elastic. In elastic collisions, the forces comprising are conservative in nature.

Suppose two objects having mass m1 and m2.

Initial velocity of object 1 = u1

Initial velocity of object 2 = u2

Suppose v1 and v2 be the final velocities correspondingly.

By applying the conservation of momentum principle, we obtain:

m1u1 + m2u2 = m1v1 + m2v2

As well applying the principle of kinetic energy conservation:

1/2 (m1 u12) + 1/2 (m2 u22) = 1/2 (m1 v12) + 1/2 (m2 v22)

By solving the above two equations for v1 and v2 we obtain:

v1 = [u1 (m1 - m2) + 2m2u2]/(m1 + m2)

Likewise,

v2 = [u2 (m2 - m1) + 2m1u1]/(m1 + m2)

Inelastic collision: This is the kind of collision in which merely momentum is conserved, not kinetic energy and is termed as inelastic collision. Most of the collisions in everyday life are inelastic in nature.

The equation which denotes the conservation of momentum is:

m1u1 + m2u2 = (m1 + m2) v

Here,

m1 = mass of object or body 1

m2 = mass of object or body 2

u1 = initial velocity of object or body 1

u2 = initial velocity of object or body 2

v = final velocity of both the objects

Final velocity is given by v = [m1u1 + m2u2]/(m1 + m2)

Loss in kinetic energy E = 1/2 (m1u22) - 1/2 (m1 + m2) v2

E = 1/2 (m1u12 + m2u22) - [(m1u1 + m2u2)/(m1 + m2)]2

More Applications (illustrations) of Newton's Law and the Conservation of Linear Momentum Law:

How Walking is Possible:

A person walks via pushing with his/her foot against the ground and the ground applies an equivalent and opposite force on the person. It is this reaction force of the ground on the person walking that moves him or her forward.

Therefore a person walking is in reality pushed forward by the reaction force of the ground on him and not through his on her own push.

Jet and Rocket Propulsion:

The principle of conservation of the linear momentum and Newton's third laws are as well in the propulsion of jet aircraft and the rockets employed for launching the satellites. Gases are burnt in the combustion chambers of the engine. Whenever jets of hot gases are expelled downwards via the tail nozzle at high speeds, from rocket or aircraft, an equivalent opposite momentum is given to the rocket or aircraft causing it to move.

Inertial Mass:

Mass was stated by Newton as the 'quantity of matter in a body'. We are familiar from the Newton's first law that 'inertia', that is, the reluctance of a body to change its state of rest or uniform motion in a straight line, is an inherent property of the matter. The mass of a body can as well be stated as 'a quantitative measure of the inertia of a body'. The more the mass a body has, the greater the force needed to change its state of or uniform motion in a straight line and the greater the force needed to give acceleration:

M ∝ F/a

Therefore, this inherent property of matter is as well termed as 'inertial mass'.

Define: The inertial mass of a body is a property of matter that represents or exhibits the resistance of the body to any type of force whatever. The mass of a body provides a quantitative measure of its inertial mass.

Weight:

Define: The weight of a body is the force acting on the body because of the reason of the earth's gravitational pull.

The weight (w) of a body is given by:

W = mg

Here, m is the mass of the body and g is the acceleration due to gravity.

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