# Lecture 11: Newton's laws

(previous, next)

## Prelude

• Discussion of signal and noise
• Discussion of what makes a good talk
• What do you remember from Friday?
• What parts of the presentation where useful?
• What parts of the presentation where useless?

## Newton's laws of motion

• A body at rest tends to stay at rest, and a body in motion tends to stay im motion, unless acted upon by forces.
• Velocity is a state.
• Acceleration times mass equals sum of forces
• $$m \frac{d^2 p}{dt^2} = \sum_{i} F_i$$
• For every action, there is an equal and opposite reaction.
• This is Newton's axiomatic approach to conservation of momentum
• Conservation of momentum is a consequence of Galilean invariance of physical systems, and is obtainable by Noether's theorem, so Newton's 3rd law is rather obsolete.

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

• Galileo, 1602, bored, chandelier swinging in Pisa
• isochronism conjecture
• period independent of mass
• Christiaan Hugens, 1656, first pendulum clock
• Tautochrome as proof that the period varies.
• Formulation: fixed rod, weight at the end.

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.