Newton’s second law

Newton’s second law (the fundamental principle of dynamics, or the fundamental relation of dynamics) designates a law of physics relating the mass of an object, and the acceleration it receives if forces are applied to it.

We can also see it as arising from the principle of virtual powers which is a dual formulation.

Newton’s second law

States

The original statement of Newton’s second law is as follows:

”Lex II: Mutationem motus proportionalem esse vi motrici impressae, et fieri secundum lineam rectam qua vis illa imprimitur.”

(”The change of momentum of a body is proportional to the impulse impressed on the body, and happens along the straight line on which that impulse is impressed.”)

In its modern version:

”In a Galilean frame of reference, the acceleration of the center of inertia of a system of constant mass m is proportional to the resultant of the forces it undergoes, and inversely proportional to m.”

This is often summed up by the equation:

a = 1/m ∑F_i

or:

∑F_i = ma

where:

• Fi denotes the external forces exerted on the object;
• m is its inertial mass (which turns out to be equal to the gravitational mass, see principle of equivalence);
• a is the acceleration of its center of inertia G;
• the term ma is sometimes called the quantity of acceleration.

Thus, the force required to accelerate an object is the product of its mass and its acceleration: the greater the mass of an object, the greater the force required to accelerate it to a determined speed (in a period of time fixed). Whatever the mass of an object, any non-zero net force applied to it produces acceleration.

(Forces can be described as a push or pull on an object. They can be due to phenomena such as gravity, magnetism, or anything that might cause a mass to accelerate.)

Theorem of momentum

If the mass does not vary over time, we can reformulate the Newton’s second law as follows: The derivative with respect to time of the momentum of a body is equal to the force applied to it.

∑F_i = dp/dt

where:

• F denotes the forces exerted on the object;
• p = mv is the momentum, equal to the product between its mass m and its speed v.

The theorem is applicable to any system of constant mass, including one formed from different pieces (subsystems). Then the momentum p is the sum of the momentum of the different subsystems:

p = ∑_jp_j.

D’Alembert’s principle

We can also write the Newton’s second law in the form:

∑F_i – ma = 0.

This allows a graphical translation of the Newton’s second law: if we put the force vectors end to end, we obtain an open polygon (since the sum of the forces is not zero); the vector –ma is the vector which closes the polygon.

We find this form by placing ourselves in the frame of reference of the studied object: if the acceleration is not zero, the frame of reference is no longer Galilean, we therefore introduce the force of inertia

F_I = − ma = −dp/dt

and we find the fundamental principle of statics (the solid being immobile in its own frame of reference)

∑F_i + ∑F_I = 0

Writing the Newton’s second law in this form makes it easier to solve some problems.

This constitutes a particular case of d’Alembert’s principle: since

F(x) – dp/dt = 0,

a fortiori

∫_C(F(x) – dp/dt)∙δr(x)dx = 0