Day 02 - Newton’s Laws

Day 02 - Newton’s Laws#

\[\mathbf{F}_{net} = m \mathbf{a}\]

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Announcements#

  • Homework 1 is due next Friday

  • Help sessions will start next week

  • Friday’s class will include AI policy discussion

    • We will work Problem 1 together and start discussing Problem 6

Goals for this week#

Be able to answer the following questions.#

  • What is Classical Mechanics?

  • How can we formulate it?

  • What are the essential physics models for single particles?

  • What mathematics do we need to get started?



Take 2 min to write down what comes to mind when asked:#


What is “Classical” Physics?#

Classical Mechanics#

Modeling large, slow-moving objects#

Newton’s Laws are but one of a number of formulations:

  • Lagrangian Mechanics

  • Hamiltonian Mechanics

  • Dynamical Systems Theory

  • …

An Overview of Different Physics#

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Classical Mechanics is still very relevant#

Tiny Limbs and Long Bodies: Coordinating Lizard Locomotion Research Lab

Tiny Limbs and Long Bodies: Coordinating Lizard Locomotion Source: https://youtu.be/Qme07fA3Fj4

Think-Pair-Share#

We used a tilted coordinate system (\(x-y\) plane) to analyze the motion of a block on an inclined plane. How can we check that we did the gravitational force decomposition correctly?

Recall:

  • \(F_{\text{gravity}},x = m g \sin(\theta)\)

  • \(F_{\text{gravity}},y = m g \cos(\theta)\)

Come up with at least two checks.

Clicker Question 2-1#

The formal definition of a Taylor series expansion around a point \(a\) is:

\[f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + \ldots\]

This formula makes me feel:

  1. Confident, I got this.

  2. A little nervous, but I think I remember.

  3. Uncomfortable, I don’t remember this.

  4. I have no idea what this is.

Think-Pair-Share#

We derived the following differential equation for the falling ball in one-dimension:

\[\frac{d^2 y}{dt^2} = +g - \frac{b}{m} \frac{dy}{dt} - \frac{c}{m} \left(\frac{dy}{dt}\right)^2\]

Let’s assume the turbulent drag term is negligible. Is there an anti-derivative of the right-hand side of this equation? If so, what is it?

\[\frac{dv}{dt} = +g - \frac{b}{m}v\]