If a body is moving without external forces, then it maintains indefinitely its
rectilinear and uniform motion (with a constant speed) or, if it is initially at rest, it
continues to be at rest .
If we consider that the greater is the inertia of a body, that is its aptitude to maintain unchanged its state of rest or rectilinear and uniform motion, the greater is its mass, that is the amount of matter it contains, then we deduce that this principle is a direct consequence of inertia, that in the international system of units is measured in kilograms.
On the Earth the effect of the gravity force can be neutralized making the bodies move on a rigid and smooth horizontal plane, to minimize the effects of the friction forces, that decelerate the motion.
To verify that a body, moving with an uniform and rectilinear motion, provided it isn't submitted to any external forces, persists in moving in a straight line with a constant speed, we can use any ivory balls,like the ones used in billiard game, making them move on several horizontal planes of different hardness and smoothness.
If we care for the billiards balls are thrown with the same speed in each test, we can be able to notice that the distance covered by them before coming to stop is the greater, the smoother is the surface.
For example,we would be able to notice that the coming-to-stop distance ,with the same starting speed, increases if the experience is executed on a horizontal marble surface, in comparison with the distance covered on a hard horizontal surface lined with cloth, that introduces a greater friction.
It can be deduced, by extrapolating the experimental results, that the rectilinear motion never would stop if it could possible to eliminate entirely all the passive resistances that decelerate the motion, that is the friction forces in the point of contact among billiard balls and surface and moreover the aerodynamic forces (the so-called motion resistance in the air).
A space-ship is the ideal laboratory to verify the inertia principle, because the gravity of the bodies moving inside it, is neutralized by the centrifugal acceleration produced by the orbital motion.
We think about the astronauts who are making mechanics experiments in gravity absence,by imparting to the bodies floating in the space-ship, little pushes that produce a rectilinear motion with a constant speed till the moment of the impact of them against the walls of the space-ship.
In the XVII century Galileo made the bodies roll on a smooth inclined plane, that allowed to him to reduce notably the value of the gravity acceleration (g = 9,8 m/s2), to perform easily his experiments.
By noticing that speed variations were always more little with decreasing of the plane inclination, he had the genial intuition to foresee that, in the case of a smooth horizontal plane,it could be achieved the indefinite maintenance of the state of rectilinear and uniform motion of a body, provided all the decelerating forces could be eliminated.
With his intuition Galileo broke definitely the connection with the
theory of Aristotle, which, without making any experiments, sentenced that a force is always needful to maintain the rectilinear and uniform motion of a body.
The impulse of a force is a vectorial physical quantity,acting in the same direction of
the force and having at any moment a modulus that is directly proportional to it.
It is definite, if the force is constant, by the product I = F t of the force for the time during which it acts on a body.
If, for example, we apply to a body a constant force of 3 kg during 10 seconds, the impulse of the force is 30 kg.sec.
The linear momentum of a body is a vectorial physical quantity,parallel to the velocity p = mv and having at any moment a modulus that is directly proportional to the mass m and the speed v of the body.
If, for example, a thirty-tons trailer truck is moving with the same speed of a one-ton car, the linear momentum of the first is thirty times greater then the one of the latter.
A force, if it is acting in the same direction of the velocity of a body, or if it is inclined with respect to the velocity by an angle which is smaller than 90°, accelerates the body.
Instead, if a force is anti-parallel to the velocity of a body, or if it is inclined with respect the velocity by an angle which is greater than 90°, it accelerates the body.
In the first case the speed and linear momentum of the body are increasing, because it is being accelerated in the same direction of its initial speed; in the second case instead, the speed and the linear momentum of the body are decreasing, because it is being decelerated in the direction of its initial speed, which is therefore being gradually reduced to zero and then, immediately,is inverted and increases in the direction of the force.
Therefore, if one or several forces, that are ever replaceable with their resultant force, act on a body, they produce as an effect the variation of the speed and linear momentum of the body.
The second law of dynamics states that, if one or several constant forces, replaceable
with their resultant force, act on a moving body during the time t ,
the impulse I of the force or of the resultant force is equal to
the variation of the linear momentum of the body, during the same time: the impulse is I = F t = P final-P initial =
m ( V final-V initial).
This fundamental law of Nature, discovered by Galileo and mathematically expressed by Newton by means of the differential calculus, that was invented, independently, by Newton and Leibnitz, is known as the second principle of dynamics, and contains, as a particular case, the inertia principle.
If, for example,we apply to a body at rest or moving on a straight line with a constant speed, a constant force of 10 kg during a time of 5 seconds, getting some increase of its linear momentum, we could verify that, if the experiment were repeated with the same body and with a constant force of 30 kg acting during the same time, the variation of the linear momentum would become three times greater than the one produced in the first test.
Since the linear momentum of a body is depending on both the speed and mass, the second law of dynamics, expressed by the variation of the momentum per a unitary time, is valid even in the particular case when the mass of a body increases or decreases during the motion, like it happens in the case of a missile, for which the mass at the moment of launching, when its tanks are full of fuel and of comburent (respectively liquid hydrogen and oxygen), is greater than the one in the final phase of the flight, when the tanks are nearly empty.
In some special cases, when the mass m of the body and the modulus of the force F are constant during the motion, the second law of dynamics may be expressed equating the force F to the product between the mass m of the body and its speed variation (V final - V initial)/(t final - t initial) per an unitary time, during the time (t final - t initial) in which the body is subjected to the force or to the resultant of the forces:
F = m (V final - V initial)/(tfinal - t initial).
In such cases, if we consider that the average acceleration a of a moving body is defined as the ratio between its speed variation
(V final - V initial) and the time (t final - t initial) we take into account, we can enunciate the second law of dynamics saying that the acceleration a of a body subjected to a force F or to the resultant of the external forces applied to it, is parallel and acts in the same direction of the force or of the resultant of the forces, and assumes a value which is in inverse-proportionality relation to the mass of the body:
a = (V final - V initial)/(tfinal - t initial) = F/m.
Therefore, with an assigned constant force F acting on a body with the mass m, the acceleration a halves if doubles the mass of the body and duplicates if instead the mass of the body halves.
By taking account of all these considerations,the second law of dynamics may be expressed saying that the force F and the acceleration a are directly proportional: F = m a or F/a = m.
To verify this fundamental law we could, for example, play by using several billiard balls of different mass: if in each test we apply always the same throwing force, we can verify that , during equal times, the speed variations are inversely proportional to the mass of the billiard balls.
This law of nature, enunciated by Galileo on the basis of experimental observations, was expressed mathematically by Newton by means of the elementary variation dP = d (MV) of the linear momentum P = MV in an elementary time dt, using the formalism of the differential calculus:
F = d ( MV ) /dt.
If we know the variation law of the force F(x,t,V) in terms of the time t, the coordinate x and the speed V , Newton's formula constitutes the differential equation of motion, that is the fundamental equation that permits to determine, by analytical methods, the variation laws of the coordinate x and the speed V of the body in terms of the time t, by knowing its initial position and speed.
If, in particular, on the body doesn't act any force, the linear momentum P = MV and the speed V are constant, because of the inertia principle.
As a further application example, we can consider the motion of a train or of a car along a curve.
The force that prevents the vehicle from going outside the curve along the direction of the tangent line in each point of it, is, in the case of the train, the centripetal force F (directed toward the center of the curve ), which is exerted by the rails;in the case of a car instead, the centripetal force F is the resultant of the friction forces (adherence forces) between the tires and the road surface.
In both cases the velocity modulus, not its direction, is constant , because the velocity, in every point of the curve,is directed along the tangent line; instead the corresponding velocity variation (and then the centripetal acceleration) is directed toward the center.
For both vehicles, the centripetal force F is equal to the product
M a = M V2/R,because the centripetal acceleration a is V2/R.
From the formula it can be deduced that, in the case of a car, by taking account of a certain least adherence between the tires and the road surface,that depends on the conditions of it (the asphalt can be dry, wet or slippery ), the smaller is the radius of the curve, therefore the narrower is the curve, the lower has to be the speed, to avoid that the available centripetal force (the friction force ), Fmin, not to be enough to maintain the motion of the vehicle along the curve.
If this condition was not satisfied, the driver would lose the control of the car, that would go outside the road along the tangent line in the point of going out.
If instead we consider the motion of a two-wheels vehicle (bicycle or motor-cycle) along a curve, to assure enough stability,it is necessary to add a further centripetal force to the one provided by the adherence between the tires and the road surface.
The additional centrifugal force is the horizontal component of the reaction force (oblique in comparison with the ground) acted by the road on the vehicle, and is generated when the driver makes the vehicle tilt of a certain angle toward the center of the curve.
In this case, the vertical component of the reaction force of the road surface equalizes the sum of the weight of the vehicle and the driver, while the horizontal component provides the additional centripetal force.
For the same reason the slope of the bicycle-racing tracks and the support plan of the railway rails have to be inclined, in a curve, by an angle which is the greater, the greater is the maximum speed along the curve.
The third law of dynamics, which is known as Newton's action-reaction principle, says
that to every action corresponds an equal and opposite reaction.
This principle states that the force a body acts on another body is always equal and opposite to the reaction force the second body acts on the first one.
We provide some application examples of this principle of Nature:
1) When we are walking, the force (action) that our feet act backwards on the ground,it is always equal and opposite to the force (reaction) acted by ground on our feet, and that permits us to move on;
2) The helix of a ship or of a helicopter effects a force (action), respectively on the water or on the air,which is equal and opposite to the force (reaction) the water or the air act on the ship respectively, making the ship move on and the helicopter balance its weight;
3) A missile acts on the heated up gases going out from the nozzles of its jet engines,with a force (action) which is equal and opposite to the force (reaction) the heated up gases act on the missile, making it fly;
4) the tires of a vehicle effect, in the contact points with the ground, forces (action forces) backwards directed,which are equal and opposite to the friction forces (reaction forces ) the ground acts on the tires, preventing the skid and making the vehicle go on;
5) A fire weapon,launching the bullet, acts on it a force (action),equal and opposite to the force (reaction) the thrown bullet effects on the fire weapon producing the recoil.
Other examples: When two bodies collide, one of them acts on the other one an action force equal and opposite to the reaction force that is acted by the other one .
The action and reaction forces act always on the same straight line and in opposite directions.
Another example of action and reaction forces is furnished by the gravitational attraction between two masses, for example between the Sun and a planet: the Sun attracts a planet with a gravitational force which is equal and opposite to the gravitational force by means of which it is attracted by the planet.
It is evident,in this case, that, because the mass of the Sun is much greater of the one of the planet, the acceleration acting on the Sun by the planet is much more little than that the one the Sun effects on the planet.
An analogous situation verifies in the case of the attractive or repulsive forces acting between electric charges.
We get another example considering the centripetal force (action force) effected by a hand tied to a body by means of a wire that makes the body move with a circular and uniform motion around the hand ,and the centrifugal force (reaction force) that the hand suffers by the body that revolves.