Physics

Dynamics I. Newton's Laws

Study of fundamental forces, linear momentum, Newton's three laws and the moment of a force.

A force, F⃗\vec{F}, is a phenomenon that modifies the motion of an object (accelerates, decelerates, changes direction, etc.) or deforms it.

Forces are vector quantities, meaning they have magnitude and direction and are represented by vectors.
Its SI unit is the Newton, N=kg⋅m⋅s−2\text{N} = \text{kg} \cdot \text{m} \cdot \text{s}^{-2}

There are 4 fundamental interactions/forces in the universe:
  • Gravitational interaction: the gravitational force is the attractive force between two bodies with mass. It always acts as an attraction from one body to another, and has an infinite range. It was described by Newton in his Law of Universal Gravitation.
  • Electromagnetic interaction: Affects bodies with electric charge, and is the force involved in the physical and chemical transformations of atoms and molecules. It is much stronger than the gravitational force, can be attractive or repulsive (positive and negative), and its range is infinite.
  • Strong nuclear force: is the force that holds the components of atomic nuclei together, and acts equally between any two nucleons, protons or neutrons. Its range is on the order of nuclear dimensions, but it is stronger than the electromagnetic force.
  • Weak nuclear force: is responsible for the beta decay of neutrons; neutrinos are sensitive only to this type of interaction. Its intensity is less than that of the electromagnetic force and its range is even smaller than that of the strong nuclear interaction.
Linear momentum, p⃗\vec{p}, is a vector quantity defined as the product of an object's mass and its velocity.
p⃗=m⋅v⃗\vec{p} = m \cdot \vec{v}
Its SI unit is kg⋅m⋅s−1\text{kg} \cdot \text{m} \cdot \text{s}^{-1}
The definition of force can be derived from the concept of linear momentum:
F⃗=dp⃗dt\vec{F} = \frac{d\vec{p}}{dt}
Principle of Conservation of Linear Momentum
If no net external force acts on an object, its linear momentum remains constant:
∑F⃗=0  ⟹  p⃗=cte\sum \vec{F} = 0 \implies \vec{p} = cte
Impulse
It is a vector quantity defined as the change in linear momentum:
I⃗=Δp⃗\vec{I} = \Delta \vec{p}
I⃗=F⃗⋅Δt\vec{I} = \vec{F} \cdot \Delta t
1st Law (Law of Inertia)
In the absence of external forces, an object at rest stays at rest, and an object in motion continues moving in a straight line with constant velocity.
∑F⃗=0  ⟹  v⃗=cte\sum \vec{F} = 0 \implies \vec{v} = cte
Inertial reference frame: is a reference frame where Newton's 1st Law holds true, meaning it is not subject to any force, rotation, or acceleration, and is therefore at rest or moving with constant velocity. The Earth is not a strictly inertial frame because it is in constant rotation.
2nd Law (Fundamental Law of Dynamics)
The acceleration acquired by an object is directly proportional to the net force acting on it, and inversely proportional to its mass.
a⃗=F⃗m\vec{a} = \frac{\vec{F}}{m}
  or  
F⃗=m⋅a⃗\vec{F} = m \cdot \vec{a}
Since F and a are vector quantities, considering a 2D x-y coordinate system, we can express the 2nd law in terms of its components:
{∑Fx=m⋅ax∑Fy=m⋅ay\begin{cases} \sum F_x = m \cdot a_x \\ \sum F_y = m \cdot a_y \end{cases}
3rd Law (Action and Reaction)
For every action, there is always an equal and opposite reaction: this means that the \mutual actions of two bodies are always equal and directed in opposite directions. It states that for every force acting on a body, the body exerts a force of equal magnitude but opposite direction on the body that produced it.
F⃗12=−F⃗21\vec{F}_{12} = -\vec{F}_{21}
The tendency or capacity of a force to rotate an object is called moment or torque. It is a vector quantity that depends on the magnitude of the applied force and the perpendicular distance from the line of action of the force to the axis of rotation. It is measured in N·m.
Moment of a force
M⃗=r⃗×F⃗\vec{M} = \vec{r} \times \vec{F}

The scalar value of torque can be calculated as:
M=F⋅dM = F \cdot d

Where d (moment arm) is the perpendicular distance from the line of action of F to the pivot point.
M=F⋅r⋅senϕM = F \cdot r \cdot \text{sen} \phi
Couple of forces
Couple (of forces): a system of two equal forces acting in opposite directions, applied at two points separated by a distance d. The torque created on the wheel will be:
M=F⋅dM = F \cdot d