Physics

Dynamics II. Fundamental Forces

Study of the main mechanical forces: gravitational force, weight, normal force, friction, and tension.

Newton's Law of Universal Gravitation: The attractive force between two bodies is equal to the product of their masses and inversely proportional to the square of the distance separating them:
F=Gm1⋅m2r2F = G \frac{m_1 \cdot m_2}{r^2}
Gravitational force
  • F: magnitude of the force. The direction is attractive along the line joining the two masses.
  • G: Universal Gravitational Constant, G=6.67⋅10−11 N ⋅m2⋅kg−2G=6.67 \cdot 10^{-11} \text{ N } \cdot \text{m}^2 \cdot \text{kg}^{-2}
  • m1_1, m2_2: masses of the bodies
  • r: distance between the centers of mass of the two bodies
Weight of an object
Weight is the force with which the Earth attracts a body. It has a vertical downward direction, applied at the center of mass of the body.
P=m⋅gP = m \cdot g

Where g is the acceleration due to gravity, g=9.81 m ⋅s−2g=9.81 \text{ m } \cdot \text{s}^{-2}
It is the reaction force that appears between the contact surfaces of two bodies when one body exerts a force on the other. It is directed perpendicularly to the supporting surface. Examples:
Normal force examples
Appears on the contact surface between two bodies when one body experiences a force that makes it tend to slide over the other. Its direction opposes the displacement of the body.
Fr=μ⋅NFr = \mu \cdot N
  • N: the normal force on the contact surface
  • μ\mu: coefficient of friction, which depends on the material of the contacting surfaces. It is dimensionless.
Friction force
A distinction is made between static friction, μS\mu_S, and dynamic (kinetic) friction, μD\mu_D:

Static coefficient of friction, μS\mu_S: applies when the resultant force pulling the body fails to move it:
FrS≤μS⋅NFr_S \leq \mu_S \cdot N
The body will begin to move when F>FrSF > Fr_S

Dynamic coefficient of friction, μD\mu_D: applies when the body slides. It is generally smaller than the static coefficient. We verify this from experience; it is harder to start dragging an object than to maintain its motion once it is moving:
FrS>FrDFr_S > Fr_D
Appears internally in a taut rope or cable. Assuming massless and frictionless ropes and pulleys, the same tension appears along the entire length of the same rope.
Tension in ropes and pulleys
Hooke's Law: The extension or compression experienced by an elastic body is directly proportional to the force applied to it.
Hooke's law in a spring
FX=k⋅xF_X = k \cdot x
  • k: proportionality constant specific to the elastic body, in N⋅m−1\text{N}\cdot\text{m}^{-1}
  • x: extension or displacement from the equilibrium position in m
A body moving in a circular path is acted upon by a force directed towards the center that keeps it on its trajectory. This is the centripetal force. This force can be the tension in the string that ties it, the friction of the road on a car in a curve, etc. If this force disappears, the body would fly off the curve along a tangential path.
FC=m⋅v2R⇒FC=m⋅ω2RF_C = m \cdot \frac{v^2}{R} \Rightarrow F_C = m \cdot \omega^2 R
Centripetal force on a curve
Centrifugal force: is a subjective, and therefore fictitious force, experienced by a body moving within a rotating frame of reference: Its magnitude would be the same as that of the centripetal force.
Coulomb's Law: The attractive or repulsive force between two electric charges is directly proportional to the product of the charges and inversely proportional to the square of the distance separating them, directed along the line joining them. The force is repulsive if the charges have the same sign, and attractive if they have opposite signs.
F=Kq1⋅q2r2F = K \frac{q_1 \cdot q_2}{r^2}
  • F: magnitude of the force.
  • K: Coulomb's constant K=9⋅109 N⋅m2⋅C−2\text{K}=9 \cdot 10^9 \text{ N}\cdot\text{m}^2\cdot\text{C}^{-2}
  • q1_1, q2_2: charges
  • r: distance between the charges