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Physics 271: Advanced Honors Physics I
Recitations
Fall 2017

Week-by-week summary of recitation activities

Week 1: df/dx=f and other separable differential equations, 1D constant acceleration review and practice (including diagnosis of need to review unit conversions and/or sig figs), diagrammatic derivation of transformation of coordinates between two coordinate systems with different origins. The sheet of practice problems and answers/comments.
Week 1 to do: finish the in-class problems (answers will be posted Friday after the last recitation). If you discovered that you are rusty on (or never properly learned) unit conversions and/or sig figs, study the links in the Week 1 Useful Resources. Do homework 1 to hand in at Monday lecture (start working on it right away if you have not already done so).

Week 2: review projectile motion, uniform circular motion, relative motion in 1D, normal force, string tension force, constraints in problems with pulleys. DO THE WARM UP PROBLEMS BEFORE RECITATION. Check your answers here: #1 #2 #3 #4 #5. If you have difficulties, review on your own or come to office hours before this week's recitation. In recitation, we reviewed the conical pendulum, talked about choosing reference frames (examples in 1D), rolling without slipping, force exerted by a string on a pulley, and practiced identifying constraints in dynamics problems.
Week 2 to do: if you have not seen it before (or even if you have), watch the short film "Frames of Reference" (links in the Useful Resources section) This post-Sputnik (1960) geeky classic is arguably the most perfect physics film ever made. To me, it's like Casablanca -- it gets better every time I watch it.

Week 3: review static and kinetic friction, drag force, simple harmonic motion, gravitational shell theorem and gravitational orbits. DO THE WARM UP PROBLEMS BEFORE RECITATION. Check your answers here: #1 #2 #3 #4 #5. If you have difficulties, review on your own or come to office hours before this week's recitation. In recitation, we reviewed drawing force diagrams in systems with static or kinetic friction, two objects of similar mass circularly orbiting around each other, and the motion of a string with mass on a frictionless pulley.
Week 3 to do: students in the Thursday 12:15 recitation should check out the rule for deciding which direction to draw the friction force which can be found in the problem solving toolbox below. Go over your notes for the motion of a string with mass on a frictionless pulley and fill in the remaining steps to find the magnitude of the acceleration of the string. If you need to refresh your memory or learn Taylor series (as used in the recitation discussion of the rope on the pulley), study Notes 1.2 and 1.3 on pp. 37-38 of KK. For a mathematical review, see the link in Useful Resources to the relevant pages (Sec 10.7 in 2nd ed) from the calculus text by Rogawski which is often used in Math 151/152/251.

Week 4: review Chapters 1-3 for in-class exam Monday October 2. An important reminder: take some time to prepare your two-sided note sheet -- this will help you study and may come in handy during the exam. We listed some of the topics that are "fair game" for the exam: reference frames (see Problem 1 of HW2), polar coordinates (you can be sure that you will need to know acceleration vector in polar coordinates!), Newton's law problems (define coordinates, constraints, drawing force diagrams (real physical forces only!), Newton's 2nd law, solving the equations, making sure they are reasonable, checking that the solution is reasonable), knowing when the friction is static and when it is kinetic (eg. "just about to slip"), remember that the static friction force is not always mu N, how to figure out the direction of the friction force when you are drawing the force diagram, know how to approach max-min problems, solve separable differential equations and determine the coefficients in the solution from initial conditions. The full list of problems that we discussed in at least one of the recitations: KK 2.15 "disk with a catch" (to review motion and acceleration in polar coordinates), KK 2.8 solution "two masses and two pulleys" (to review how to write constraint equations), KK 2.12 "painter on scaffold" (to review force diagrams and Newton's THIRD Law), and KK 3.17 "turning car" (circular motion and friction). Scanned solutions for these 4 problems coming...

Week 5: review center of mass, momentum, conservation of momentum, impulse, rocket motion. DO THE WARM UP PROBLEMS BEFORE RECITATION. Check your answers here: #5 #11 #25 #39 #77 #79. We discussed Problem 4.5 (acrobat and monkey), rederived the rocket equation keeping open the possibility of various M(t) and external forces on the rocket, derived a similar general equation for freight cars and sand and applied it to various freight-car-and-sand problems. YOU ARE READY AND STRONGLY ENCOURAGED TO DO THE FOLLOWING 7 PROBLEMS FROM HW4 FOR THIS MONDAY (10/9): 4.1, 4.4, 4.7, 4.9, 4.10, 4.15, 4.16. A few helpful comments and pointers, most of which we discussed in recitation, are as follows:
KK 4.4. Treat the rocket as a projectile that's launched at a particular speed and angle from the launch point.
KK 4.7 Divide the motion into two time intervals and solve in each time interval separately using an appropriate approach.
KK 4.7 You will need to understand the motion of the center of mass (though in fact not the motions of the individual masses, though it is good more generally for you to understand those too) in the Push-Me-Pull-You in Example 4.7. A link to a video demo of the Push-Me-Pull-You is in the Useful Resources below.
KK 4.9 Take the sled's initial mass as M_0 (a notation that will reduce confusion with the function M(t))
KK 4.16 I suggest you solve this problem using the techniques from Chapter 3 (that is, don't try to use momentum, as the external forces exerted by the corner of the table on the string are not obvious).

Week 6: review flow of momentum, work done by various forces in 1D, 2D and 3D, work-kinetic energy theorem. DO THE WARM UP PROBLEMS BEFORE RECITATION. Check your answers here: #29 #8 #17 #27 #37 #39.

Week 7: review power, small oscillations about equilibrium and elastic and inelastic collisions in 1D and 2D. DO THE WARM UP PROBLEMS BEFORE RECITATION. Check your answers here: #45 #11 #83 #50 #61 #71.
IMPORTANT: in inelastic collisions, momentum is conserved but mechanical energy is NOT! The force exerted on object A by object B is equal and opposite to the force exerted on object B by object A (Newton's 3rd law) but the work done by object B on object A is not necessarily equal and opposite to the work done by object A on object B for nonconservative forces such as friction.

Week 8: review of moments of inertia, rotational kinetic energy, torque, work done by torque, conservation of angular momentum. DO THE WARM UP PROBLEMS BEFORE RECITATION. Check your answers here: #35 #45 #59 #47 #53 #7.
The geometry of the bug and ring problem in the HW is that the ring is lying flat on the table (x-y coordinate system). It has a hole drilled for the pivot and rotates around the pivot point: define the angle of the ring's position to be the angle made with the x axis by the vector from the pivot to the center of the ring.
Tip: for the falling stick problem in 7.14, the force applied at A is only to establish the initial condition for the falling stick -- that it is horizontal and not moving at t=0.
Tip: for the small oscillation problems 7.16 and 7.17, you can either compute the restoring torque for a small displacement OR expand the potential energy to quadratic order in the small displacement amplitude. For the small oscillation problem 7.9, you have no choice because of the friction forces -- you have to compute the restoring force for a small displacement.
In Problem 7.9, treat the bar as being infinitesimally thin.

Week 9: review of rolling without slipping, conservation of energy in rolling problems, the yo-yo. DO THE WARM UP PROBLEMS BEFORE RECITATION. Check your answers here: #2 #3 #11 #18 #51 #83.
*Most of the HW problems are best tackled by the extended Newton's law approach. First draw a diagram and identify the forces. Write the constraints (eg the rolling without slipping condition), Newton's 2nd law for the center of mass motion, and the torque equation, and solve. The problems involving collisions need to be solved using conservation laws (momentum, angular momentum, and mechanical energy -- it's up to you to recognize which quantities are conserved in each problem) -- the forces acting during the collision are too complicated, which rules out the Newton's law approach. For conservation law problems, draw detailed before and after diagrams.
*7.34 is another small oscillation problem (see the tip for week 8). All small oscillations (of which we have already done many and will do many more) should be approached the same way. Draw a diagram of the equilibrium state and identify the "small displacement" quantity (it might be an angle). Then, if the forces in the problem are conservative, you have a choice: you can either compute the restoring force (or torque if the small displacement is an angle) and put it into a form proportional to the small displacement quantity (this might involve a linear approximation to a function of the quantity) OR you can write the kinetic and potential energies, and expand the potential energy to QUADRATIC order in the small displacement quantity). In the force/torque case, then look at your Newton's law/torque equation and map the quantities to the spring equation m x.. = - k x to get the frequency f = root (k/m)/(2 pi). In the energy case, look at the mechanical energy and map the quantities to the spring mechanical energy (1/2) m x.^2 + (1/2) k x^2 to get the frequency f = root (k/m)/(2 pi). NOTE that if there are nonconservative forces (eg kinetic friction) in the problem then you HAVE TO compute the restoring force or torque (example is the bar and rollers problem from HW 7).
*Specifically for 7.34: you are given that b<<R. This means that you are allowed to replace R - b by R and that you are allowed to use the flat surface rolling without slipping condition for the curved surface of the dish, which makes the problem quite easy. In fact, with a little more care, you can solve the problem without using approximations based on b<<R and still get a simple answer which is not the same but becomes the same in the limit b/R -> 0. If you want to try this, let me know and I'll give you any help you need.
*The optional falling plank problem is not as difficult as the book scares you into thinking. You should have no difficulty writing the equations with the extended Newton's law approach. It is true that to solve the equations, you need a small "trick" to get there quickly. If you are having trouble solving your equations, come to office hours or email them to me (ok to email a photo from your phone). I will check them and if correct, tell you the trick so you can finish the problem easily.
*The optional two rubber wheels problem is a rolling with slipping problem like 7.30 and 7.31. The angular momentum of the two-wheel system is NOT conserved -- can you see why?

Week 10: Exam review. List of topics covered on the exam here. We discussed strategies for recognizing and getting started with the different types of problems that might appear on the exam, and how to prepare your formula sheet to help with this. The specific types of problems we discussed were drawn from the practice exam: collision problems (#9), conservation of energy problems (#9) and rocket problems (#11). Many times the problems will be combinations of types, for example collision with fixed axis rotation (#12).

Week 11: Rigid body motion. First, a general strategy for solving the problems:
Step 1: find fixed point (need not be in the body). Write vector omega using the description of the motion given in the problem.
Step 2: Find vector L. For most of the problems, I recommend finding L with respect to the center of mass (the vector omega is the same). Decompose vector omega into pieces for which the angular momentum is parallel to the angular velocity,  Here's a big time-saver - in most of the problems, the next step will be to differentiate L with respect to t to get the torque, so you only need to find the component of L that depends on time -- don't waste effort working out the time-independent component. 
Step 3: Compute vector dL/dt.
Step 4: Draw a force diagram showing all forces and points of application. Find the torque around the reference point you chose in step 2. Set this equal to dL/dt from step 3. Solve the equation for the unknown quantity,
Now, some comments on the specifics of the problems:
8.2. In the exact solution, the rotation of the disk by the turntable contributes a time-dependent angular momentum when the disk is tilted. Here, the statement that Capital Omega << omega_0 means that you are allowed to neglect this. So, the only time-dependent angular momentum that you need to consider comes from the omega_0 spin of the disk.
8.4. In order for us to compute the forces and torques on the system, since we haven't discussed the torque when the contact is not a point, we need to redraw the mechanism shown. Neglect w and imagine that the axle is rigidly attached to the wheel and connected to the shaft by a ball and socket arrangement that allows free rotation around the axis defined by the axle and is fixed in the other directions (stays perpendicular to the shaft and rotates with it). Then you can identify the force exerted by the shaft on the rigid body consisting of the wheel and axle.
8.6. (1) NOTE: the picture has many errors -- the labelling of b and R and the direction of omega_s, in particular, so draw your own.
(2) The coin cannot roll in a circle without static friction between the coin and the floor. Look at the motion of the cm and use Newton's 2nd law to help you find the direction and magnitude of the friction force.
(3) Assume b<<R. This allows you to make the approximation in the parentheses, and also to neglect the contribution to the angular momentum of the rotation of the coin around its diameter as it rolls (this makes it a lot easier to solve the final equation).

Week 12: Fictitious forces. Here's a alternative derivation of the expression for the fictitious force in a rotating frame (this is what we discussed in the Thursday office hour) here.
In the Friday recitation, we considered what the motion of an object that is stationary in the inertial frame looks like in the noninertial frame, and related that motion to the ficitious force we computed for the noninertial frame. I'll add a writeup here if anyone asks for it.
The general strategy for solving the problems:
Step 1: Identify the noninertial frame you are working and compute the fictitious force.
Step 2: Identify the real forces acting.
Step 3. Solve the problem (usually using Newton's 2nd law) including both real and fictitious forces.
Now, some comments on the specifics of the problems:
9.1. At equilibrium, this rod is balanced on the pivot. The equilibrium is UNSTABLE (like a ball at the top of a hill) -- if the rod is displaced by a small angle or given a small angular velocity at the equilbrium position, it will not return to equilibrium -- it will fall. Note that the differential equation describing the motion is the same type as that which describes the motion of the bead on the wire.
9.7, 9.9, 9.11. You will need to be careful to draw vector omega and vector r correctly in the reference frame fixed to the surface of the earth. Vector r is always pointing up for an observer standing on the surface of the earth. The direction of vector omega depends on your latitude. For example, if you are at the North Pole then vector omega points up.
9.12 First find the equilibrium angle at which the pendulum hangs. Then use Taylor expansion to expand the torque as a function of the small angle that the pendulum is displaced away from equilibrium (it might be helpful to re-read the small oscillation problem tips from week 9 above).


Week 13: Central forces. DO THE WARM UP PROBLEMS BEFORE RECITATION. Check your answers here: #45 #49 #51 #63.
ellipse facts
short excerpt from HRW10 on Kepler's laws
(see fig 13-12 for quick basic ellipse facts)

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Useful resources

Office hours

Referring back to your previous physics textbook

In the first recitation, I suggested that you could use the textbook from your previous physics course as a reference to refresh your memory on basics you have learned before which we expect you to know. I did not take into account the fact that in most high school courses, you give the textbook back to the school at the end of the course. There are of course lots of resources on the internet where you can find the materials you need. However, if you would find it helpful to have a standard university-level calculus-based textbook (maybe even the same one you used) on hand for reference, I can lend you one for the duration of the course -- send me an email (kmrabe@physics.rutgers.edu) to let me know if you are interested. Depending on how many people ask, I should be able to get one to you without much difficulty.

If you have questions about the course or about the homework problems, email me at kmrabe@physics.rutgers.edu


This page is maintained by Karin Rabe.