DYNAMICS 

Dynamics of a particle 

The laws of motion formulated by Newton are the pillars that establish the cause (Force) and the effect (Acceleration) relation. These causal relations provide for the analysis and prediction of motion. Newton formulated the laws of motion based on Galileo's experiments dealing with rolling bodies within inclined planes. These laws are abstractions or idealizations of the experimental observations.

Newton's Laws of Motion

First Law

An object at rest or in uniform motion in a straight line continues to maintain the state of rest or uniform motion unless an unbalanced external force is applied to it. This law essentially defines force as the cause of change in a state of rest or of uniform motion in a straight line i.e. acceleration. It also defines a property of matter called Inertia as that property, which does not let the object acquire acceleration by itself.

Note : Uniform motion in a straight line means constant velocity.

That part of the statement of the first law about objects at rest is a factual statement. But the part concerning uniform motion in a straight line is an idealization of actual observations; the idealization demands that roads are perfectly smooth, leveled in horizontal plane and literally infinite in extent. Surely such a state of affairs can never be realized. The important point to be noted is that force is the cause of acceleration, whether the acceleration is due to a change in the state of uniform motion or is due to change in state of rest. It must have occurred to you that the state of rest and the state of uniform motion are equivalent; the difference lies in the frame of reference chosen for determining the state of motion.

The measure of the inertia of an object is the mass of the object. What is the mass of an object ? It is dangerous to try and define what is mass, therefore we shall not define it, but agree that all of us initially understand what it means. Of course, we will explain the method to measure it, but for that we shall have to wait till the second law.

The other name for the First law is the Law of Inertia, as it defines the property of Inertia.

Second Law

It is understood that it is an unbalanced, external force.

[The choice of unit of force is 1n - 1 Newton -when m = 1 kg and a = 1 m/s²] 

Hence, 'Force equals rate of change of momentum' is an alternate form of the second law.

Further,
Therefore, 'Impulse of the force equals change in momentum' is another alternate form of the second law.

Note  

For a particle, equations (1), (2) and (3) are completely equivalent; but for a system with varying mass and for pedagogic reasons, equation (2) is the appropriate form of the law.

Note
In scalar form

F = ma

Thus, the ratio of the force applied to the acceleration produced gives us the mass of the object.

Third Law

If an object A exerts a force (called action) on another object B, then the object B exerts equal and opposite force (called Reaction) on object A.

Note

1. The terms action and reaction are mutually interchangeable. i.e. force by B on A is action and force by A on B (equal and opposite) reaction.
2. The action and reaction occur simultaneously.
3. The action and reaction can never act on the same object.

Friction and Force of Friction

The opposition to the motion of an object moving over the surface of another object, is called friction; the force arising due to friction is called the force of friction.

The force of friction opposing the motion of an object from rest is called static friction; and the force of friction opposing the moving object is called Kinetic friction.

These forces of friction possess the following characteristics :

1. They always act in the opposite direction to motion.
2. They increase in magnitude upto a certain maximum value. This maximum force is called limiting the force of friction.
3. The limiting force of friction is independent of the area of contact between the surfaces.
4. The limiting force of friction is proportional to the normal reaction force.
5. For the same object, the limiting force of friction is different for different surfaces.
6. For the same object, the limiting force of friction is different for the same surface, depending upon the lubrication of the surface.

Newton's law of Universal gravitation

Newton discovered this law, in the process of analyzing and interpreting Kepler's laws of planetary motions and realized that the same law provides explanation for the revolution of the moon around the Earth as well as that of falling bodies on the Earth. Newton's law states:

"Every object exerts an attractive gravitational force on every other object in the universe; the force is proportional to each mass and inversely proportional to the square of distance between them."

Centripetal force in uniform circular motion

The centripetal force is a generic term. In practice the real interactions such as,

1. gravitational (in case of planets around the sun or artificial satellites or the moon around the Earth).
2. Coulomb interaction (between nucleus and electrons in the atom)

serve as the centripetal force.

Inertial and Non-inertial Frames of References

A frame of reference in which the law of inertia holds good is called as the inertial frame of reference. It is found that is the frames of references in relative translational motion with uniform velocity, are inertial frames if atleast one of them can be considered as an inertial frame.

When we travel in a vehicle moving with uniform velocity we do not experience any jerks, because the vehicle's frame of reference is an inertial frame. Of course the `ground' as a frame of reference is assumed to be in the inertial frame. In such frames, the analysis of motion can be done by using Newton's laws of motion.

A frame of reference in which the law of inertia does not hold true is called a non-inertial frame of reference. Any accelerated frame is considered to be a non-inertial frame. In such frames Newton's laws of motion cannot be used to analyze motion unless imaginary or pseudo forces are assumed to act on every object.

If the vehicle in which we are traveling speeds up or slows down though it continues to move on a straight path, we experience jerks (pseudo force effect) backwards or forwards. Likewise, when the vehicle swerves, we experience a push (pseudo force effect) in the radially outward direction called Centrifugal force. It should be noted that the experience of jerks is not psuedo. Its interpretation in terms of Newton's laws is called a pseudo-force effect, because these forces cannot be traced to any real physical interaction between the object and its surroundings.

Centrifugal force

     

  

It should be noted that centrifugal force acts (is required to act) relative to the frame E' only where as the real interaction force will act independently then it serves the purpose as a centripetal force and is responsible for u.c.m. of 'm', but if E' is chosen then it is balanced by centrifugal force and `m' remains in equilibrium relative to E' .

The pushes experienced in swerving vehicles, merry-go-round etc. are the effects of centrifugal forces; also the effects of changing weight and associated fear ; nausea etc. while riding on giant wheels, roller coasters etc. are also the effects of the centrifugal forces. 

Dynamics of a system of particles

Center of mass of a system

Newtonian mechanics is based on laws extracted from real objects of finite extensions but abstracted to formulation in terms of a particle; and as such would be of little use unless it can be extended to explain the mechanics of real objects, which can be considered as a system of either discrete particles or continuous distribution of matter.

In principle one would have to solve equations of motion for all the particles as are present and that would be an impossible proposition. However additional concepts such as center of mass if introduced, greatly simplify the problem.

Definition of Center of Mass

The computations for C.M. are usually tedious, but when the distribution of matter is symmetric about any axis, then the C.M. must lie on that axis; If more than one intersecting axes of symmetries exist than the point of intersection must be the centre of mass. This is obvious consequence of C.M. lying on the axis of symmetry.

Thus the C.M. of ring, disc, rod, cylinder, sphere etc. are their respective geometric centers.

Equivalence of system to C.M. in case of translational motion

For K^th particle in the system,

  

 

Thus, the system in translational motion is equivalent to a point mass (equal to the whole of mass of the system) located at the C.M. This is the reason why the whole mass of the system is considered as being concentrated at the C.M.

From eqn (6) we get,
 




Law of Conservation of momentum

From eqn. (6)

Equation (7) is called the Law of conservation of momentum. The total linear momentum of a system on which no external force acts remains conserved, if no external force acts on it.

Solved Problem

1. Find the minimum force with which an object of mass 2 kg should be pressed against a rough vertical wall if the coefficient of static riction between the object and the wall is 0.5. If a force greater than this minimum force is exerted will the object rise up along the wall ? Explain. (g = 9.8 m/s²)

Solution
P : Applied force.
f : Limiting (Maximum) force of static friction.
R : Normal Reaction by the wall on the object.

For Equilibrium, f = Mg & R = P

  

2. The moon revolves around the earth in (assumed) circular orbit. Draw neat labelled diagram and indicate the following in the diagram.
   (i) centripetal force

   

   .
   (ii) centri-fugal force.
   (iii) centri-fugal reaction.
   (iv) Inertial frame of reference.
   (v) Non-inertial frame of reference.

3. Estimate the mass of the sun if the earth revolves in an (assumed)circular orbit of radius 150 million kilometers around the sun. Take the period of the earth's revolution to be 365 days and G=6.67 x 10^-11 N x m² / Kg²

Solution