Lecture 11: Newton's laws

(previous, next)

Prelude

Newton's laws of motion

To solve these equations numerically, observe that second-order ordinary differential equations can be re-written as a system of two first-order differential equations. Let \(u(t) = y(t)\) and \(v(t) = \dot{y}(t)\). Then \[\begin{align} \dot{u} &= v \\ \dot{v} &= F \end{align}\]

Ballistic motion

The simplest application of Newton's laws is perhaps that of ballistic motion of a stone thrown into the air. (Or Newton's apple, in story terms.) Gravity exhurts an approximately constant force on all objects, proportional to their mass and downward. \[m \frac{d^2y}{dt^2} = \sum_{i} F_i\] \[m \ddot{y} = m g, \quad \ddot{y} = g, \quad y(t) = \frac{g t^2}{2} + \dot{y}(0) t + y(0)\] where \(g = 9.8 \; \text{m}/\text{s}^2\) is the gravitational accelaration on earth.

Pendulum

There are atleast 4 ways to derive the equations for a pendulum's motion.

  1. arclength force balance
  2. derivation by conservation of energy
  3. derivation by fictious force
  4. allow for linear spring pendulum - New set of equations!

Show small-angle asymptotics for period approximation.

Pendulum Derivations

What follows are 3 different derivations of the simple pendulum's governing equation.

Arclength derivation

A pendulum swings along a portion of a circular arc. The length of this arc \(s = \theta r\) where \(r\) is the pendulum length and \(\theta\) is the angle at a given time. Gravity exhurts a force to shrink arclength. The gravitational force is \(mg\), so the component of the force parallel to change in arclength is \(m g \sin \theta\). By Newton's law \(F=ma\), where \(a\) is the acceleration of the change in arclength. Substituting, \[- mg \sin \theta = m \ddot{s} = m r \ddot{\theta},\] so \[\ddot{\theta} = -\frac{g}{r} \sin \theta.\]

But this derivation requires one to have faith that Newton's law applies to the curved motion of \(s\) just as it does to straight-line motion.