Newton's law Problems

The problems are from Chapter 3 of Morin.

• 3.2: Double Atwood’s machine: Prequel for the next problem

• 3.3: Infinite Atwood’s machine: So far the best problem. It teaches on how to find physically equivalent situations that simplify a problem, and we also have to find $4^N/{\sum }_{i=0}^N=4^i$.

• 3.6: Sliding down the plane: This problem shows the difference between acceleration along the plane (gsin) and horizontal acceleration (gsincos).

• 3.8: Great problem on two sliding blocks. My first mistake was to assume $N=mg\cos (\theta )$, recall why it’s wrong. The key idea is to find what sliding with contact means $\to$ you have to switch frames.

• 3.9: Just double integration

• 3.11: Falling chain: If a chain remains intact, every part of it must gain the same speed! $\rho yg=\rho l\ddot{y}$

• 3.13: Balancing a pencil: The best problem so far! Solving differential equation (use the fact that it’s linear), and using Heisenberg’s uncertainty principle. Amazing final result!

• 3.17: Ball thrown from height h: The result is interesting (symmetrical answer), but it requires too much math grinding (factor when possible!).

• 3.18: Relies on 3.17 Answer. I got stuck because of mismatched convention from 3.17 … An important concept from 3.17 & 3.18 is to imagine the path on object in a reverse case (projectile motion). This can simplify the problem

• 3.19: Longest Path Integral. My main mistake was overseeing a simpler integral. Recall that $L={\int }_0^t||c^{{}^{\prime }}(t)||dt$. Other part is just math and ways to integrate $\int \sqrt{1+x^2}dx$

• 3.22: Instructive on Newton in Polar coordinates. Try to write the equations as $m{v}^2/r,m\dot{v}$.

• 3.24: The easier version of 3.25 (2/4 integrations).

• 3.25: Intuitively, answer is obvious: $\cos (\theta )=r_0/r$ for a linear motion in polar coordinates. You can derive it with Newton’s law doing 4 integrations (personally, some of them are sus because $dr/dt\cdot 1/r\cdot dt=dr/r$)

• 3.30

• 3.32

• 3.33

• 3.36

• 3.44: Newton’s apple: the proof you’ll never forget.

• 3.47: Hit the wall.

• 3.50

• 3.53

• 3.71: In a circular motion, you can write $F_r=m{v}^2/r$, $F_t=mr\ddot{\theta }=ma$