# The second law of dynamics

The second law of dynamics

In accordance with the principle of inertia, if any body, stationary (or moving at constant speed) on a perfectly smooth surface and completely free to move in all horizontal directions, a force is applied (for simplicity supposedly parallel to the plane d ‘support), its speed varies. It is possible to build in the laboratory with a sufficient approximation an “ideal” environment where all possible interferences on the movement of a body (friction) are minimized.

Consider, for example, a smooth disc that runs on a sheet of ice, pulled with a rope to whose opposite end constant force is applied, and suppose you photograph the scene in sequence, by means of shots at regular time intervals and always exposing the same stretch of film: a succession of images of the disc will be obtained, separated by increasing distances, such as to satisfy the law of uniformly accelerated rectilinear motion. The disc moves constantly accelerating, in the same direction of application of the force. Every time the experience is repeated, starting from the same initial conditions, the same final result will be obtained, with an identical constant acceleration value. Also, doubling, tripling etc. the force impressed, also the value of the

Now let’s repeat the experiment with bodies made of the same material, but of increasing size, for example with iron balls of different sizes. It will be noticed that the more the “quantity of matter” of the bodies increases (in our case the quantity of iron) the greater the resistance, or inertia, that they oppose to the force: as a consequence, the lesser the accelerations will be equal, with the same force applied, they suffer. We can then introduce a new quantity , the mass (symbol m ) which represents the measure of inertia – and therefore an indicator of the quantity of matter – of a body (for this reason it is also called inertial mass). We will say then that two bodies, subjected to the same force F , assume equal acceleration, a ,1and m2they are equal, while the accelerations are, for example, one double, triple etc. the other when the body with greater acceleration has mass exactly equal to half, one third, etc. other.

In other words, it appears that the acceleration imparted to a body of known mass m is inversely proportional to its mass and directly proportional to the intensity of the action to which it is subjected, that is:

Turning to vector notation, we can enunciate the second law of dynamics (or fundamental principle of dynamics) which establishes that, when a force F is applied to a body of mass m , it acquires an acceleration a , with directions and directions coinciding with the strength, such that:

Unit of measure for mass and force

In the International System, the unit of measurement of mass constitutes one of the seven fundamental units from which all the others are subsequently derived. This means that to define the unit of mass, you have to choose a sample object to which arbitrarily assign a unit value. At the International Bureau of Weights and Measures in Sèvres, France, a platinum-iridium cylinder is kept (an alloy that ensures a certain immutability over time), which represents the standard unit of measure of mass in the International System; it is called kilogram (also kilogram-mass) and indicated with the symbol kg (or kgm).

For the second law of dynamics, the force module has the size of a mass for acceleration. By measuring the mass in kilograms (kg) and the acceleration in meters per second squared (m / s 2 ), the unit of measurement of the force will be expressed in kg · m / s 2 . This unit is called newton (symbol N): it will have an intensity of 1 N that force which, applied to a body of mass 1 kg, causes an acceleration of 1 m / s 2 . The force measuring instrument is the dynamometer .

Weight

The application of the second law of dynamics allows the definition of a particular force, to which all bodies are subject, at least as long as they remain on the earth’s surface or in its immediate vicinity: the force of gravity . This force (which is only attractive) is exerted mutually by all the bodies of the Universe endowed with a mass. If we consider a body of mass m , first held suspended at a certain height and then dropped to the ground, it moves under the action of a force that draws it to the center of the Earth. According to the second law of dynamics, if we indicate with g the acceleration with which the body is attracted to the ground, the force, indicated with P , is given by:

where g is called gravitational acceleration and is represented by a vector directed downwards, whose modulus, as can be deduced with a succession of measurements, varies slightly depending on the place of the experiment (in particular, g assumes maximum value ai poles and minimum at the equator and also decreases with the distance from the planet’s surface.On average, g has an intensity equal to 9.8062 m / s 2 ). The acceleration of gravity, in ideal conditions, that is, in the absence of friction and measured in the same place, it is constant for all bodies. This fact is apparently surprising, because acceleration would be expected to vary according to the mass of the object, but this is not the case. This can be verified by using a tube in which the vacuum has been made and by dropping objects of different sizes and masses (the case of a ball and a feather is classic): by inverting the tube, the objects arrive at its lower end all in the same instant, since the air resistance was canceled. The law that describes the motion of an object in free fall is that of uniformly accelerated motion and it can be said that the space s traveled by the body subjected to the acceleration of gravityg in a time t is given by:

The force vector P is called the weight force or, more simply, the weight of the object under examination. Sometimes the concept of weight is confused with the concept of mass: it should be emphasized that mass is a quantity that, in addition to having a scalar and non-vector nature, has a physical meaning very different from that of weight, even if the common daily lexicon tends to use the two terms indifferently. In the International System the weight, being a force, is measured in newtons, but for practical purposes the kilogram-weight is also used as a unit of measurement (symbol kgp), where 1 kgp = 9.81 N.