Theoretical Mechanics: Fall-2021
HW 8 : Due

1. Braking Disk S0 4097S A disk with mass \(m\) and radius \(R\) is spinning with angular velocity \(\omega_0\).
   1. How much friction force would you need to apply to the rim in order to bring a disk to a halt in time \(T_0\)?
   2. Sensemaking: Check Dimensions First, indicate the dimensions you think your answer should have. Then, check that you answer has the expected dimensions.
   3. Sensemaking: Evaluate the Functional Behavior First, indicate how you expect the friction force to vary if you varied the initial angular velocity and the stopping time. Then, check that your answer has the expected functional behavior.
2. Linear Air Drag Power Series S0 4097S

   For a projectile with linear drag, the vertical component of the position looks like:

   \[y(t) = \frac{m}{b}\Big(v_{y,0} + \frac{mg}{b} \Big)\Big(1-e^{-bt/m}\Big) - \Big(\frac{mg}{b}\Big)t + y_0 \]

   1. Approximate with a Power Series: Do a power series expansion of the vertical component of the position to third order with respect to the constant \(b\).

   2. Consider a Special Case: Show that if the air drag goes to zero, your expansion gives you the equation for the vertical component of the position if there is no air drag.

3. Cycle Racer S0 4097S

   (modified from Taylor 2.25)

   Consider a cyclist moving in a straight line and coasting to a stop under the influence of a quadratic air resistance \(\vec{F}_{drag}=-cv^2 \hat{v}\).

   1. Anticipate the functional behavior: Before doing any calculation, sketch (by hand) what you expect the velocity vs time and the position vs. time graph to look like. Label interesting regions or points.
      

   2. Calculate: Starting with Newton's 2nd Law, find the velocity and position of the cyclist as a function of time.
      

   3. Dimensions: Check the dimensions of the answers.
      

   4. Examine the Behavior of Functions: Plot the velocity and position as functions of time. Do your plots make conceptual sense? Compare your plots to the your hand-sketches and comment on similarities and differences.

4. Quadratic Drag with Gravity S0 4097S

   (modified from Taylor 2.35)

   Consider an object dropped near the surface of Earth and subject to a quadratic drag force from the air.

   1. Anticipate the functional behavior: Before doing any calculation, sketch (by hand) what you expect the velocity vs time and the position vs. time graph to look like. Label interesting regions or points.
      

   2. Calculate: Find the velocity as a function of time (the equation is in the textbook but I want to see the details of the calculations). Be sure to do any integrals involved.

   3. Examine the Behavior of Functions: Plot the velocity and explain the features of this plot. Compare your plot to your hand-sketch and comment on similarities and differences.

   4. Calculate: Find the position as a function of time (again, the equation is in the textbook but I want to see the details of the calculation). Be sure to do any integrals involved.

   5. Examine the Behavior of Functions: Plot the position and explain the features of this plot. Compare your plot to your hand-sketch and comment on similarities and differences.