# Newton's First and Second Laws

As kinematics doesn't study the causes of motion, we have to introduce another branch of physics, that we call dynamics, to study it. Dynamics wants to answer the question: why does motion occur?

## First principle of dynamics (or Newton's first law)

The first scientist who studied the causes of motion was Galileo Galilei, who made experiments of motion on a slope. He saw that the speed of marble launched uphill progressively reduced, on the other hand the speed of a marble launched downhill progressively increased and this accelerations were proportional to the grade of the slope. So he made this hypothesis: without a slope, the speed of the marble is constant, so it moves of uniform motion on a straight line. obviously this could be only a mental experiment, but with this Galileo understood that if we could take away the friction of the marble with the surface of motion and with air, it would move forever. If the marble stops, then there should be an external force which makes the marble slow down. The first principle of dynamics is made up of this hypothesis.

Definition (First principle of dynamics (or Newton's first law))

A body on which no force acts remains in his state of motion, either a uniform motion on a straight line or rest

We have now considered a body on which no force acts, but what about a body on which are applied more than one force? In this case, which is the most general and correct, the Newton's first law has another definition.

Definition (First principle of dynamic's (or Newton's first law))

A body on which are applied two or more forces whose resultant is null remains in his state of motion, either a uniform motion on a straight line or rest

But now a question rises: what is a force? It could be a push or a traction but not only this. A force is defined as the physical quantity which expresses and measures the interaction between physics systems. A force is a vectorial quantity, so it has a magnitude and a direction. How can we measure a force? We can use a dynamometer, an instrument composed of a calibrated spring.

## Second principle of dynamics (or Newton's second law)

Let's now attach a spring to a body with a certain mass. If we apply a force which stretches the spring, we will detect an acceleration $a_{0}$ . Let's now repeat the experiment with a body with a different mass; applying the same force we will now detect another acceleration $a_{1}$ . If we repeat the experiment another time applying a different force, the accelerations of the two bodies will be now $a'_{0}$ and $a'_{1}$ . If we now consider the fractions

${\frac {a_{0}}{a_{1}}}{\text{ and }}{\frac {a'_{0}}{a'_{1}}}$ we can observe that they are equal. This implies that there should be an intrinsic property of bodies. We call that property mass. We can now define the second law of dynamics.

Definition (Second principle of dynamics (or Newton's second law))

The acceleration of an object as produced by a force is directly proportional to the magnitude of the net force, in the same direction as the force, and inversely proportional to the mass of the object.

This definition is then traduced in the equation

$a={\frac {F}{m}}\rightarrow F=ma$ . As we can see from the equation, the bigger is the mass, the smallest is the acceleration when the applied force is the same. Due to this property, the mass is called inertial mass.

The mesure unit of the force is the Newton [N], which is so defined.

Definition (of 1 Newton)

1 Newton is defined as the force which is necessary to accelerate a mass of 1Kg of 1m/s^{2}. So

$[N]={\frac {[Kg][m]}{[s]^{2}}}$ If on the body are applied two or more forces we have to sum them up, so the second law of dynamics will be

$\sum _{i=1}^{n}{\vec {F}}_{i}=m{\vec {a}}$ 