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PHYS 206 Problem Set 1
Instructions:
There are 2 questions. Answer each question separately, not all in one document.
The problem sets are meant to highlight the physics and methodology to solving problems. We do not just want the answer, but we want explanations for how you can obtain the answer. See the textbook for examples. We will use the following rubric to evaluate the problem sets:
1. 8-10 - Excellent (your solution is correct or has only a very small mistake, e.g. due to inat-tention, the physics is well-documented and justified throughout, constraints are thoroughly outlined and considered, and the solution applies thoughtful analysis.)
2. 6-8 - Good (your solution is correct or has very small mistakes, the physics is explained at all major steps, but perhaps not fully, some constraints and some evidence-based analysis are considered.)
3. 5-6 - Satisfactory (your solution could be correct, but you did not justify it well, or you made a small conceptual mistake without comment, e.g., overall you have the right idea)
4. 2-5 - Developing (your solution contains significant conceptual mistakes, but you made an attempt to understand the problem and displayed professionalism in your answer).
5. 0-2 - Incomplete (you did not hand in the solution, you showed little to no work, your solution is not related to the problem being asked, or there is an obvious lack of professionalism, e.g., Deviation from Academic Integrity)
The problem set must be completed in your own words. You can work in groups or with the department tutors, but you must write up your solutions on your own and in your unique voice. Students that submit identical work will receive a zero, regardless of who actually did the work. Do not upload questions to the internet. Penalties for uploading problem set questions to crowdsourcing websites, homework cheating websites, or AI applications include but are not limited to an automatic zero on the assignment, a zero in their professionalism grade, and a Departure from Academic Integrity investigation.
Submission:
If you write out the solutions, you will be using Crowdmark to submit your answers. For pencil and paper solutions, scan or take a photograph of each question and save them as separate files. You are responsible for making sure your solutions are legible. Make sure your files are in PNG, PDF, or JPG format to upload them to Crowdmark. You should receive a link for the course Crowdmark before the problem set is due. Check your spam folders for this link. If you do not receive this link, then contact Prof Sadavoy.
If you use an alternate solution method (e.g., you create a video), upload your file to the appropriate dropbox on onQ with each file labeled as LASTNAME FIRSTNAME PS# Q#, where # indicates the problem set number and question number. Again, upload a separate file for each question. In addition, upload a note for each question to Crowdmark directing the marker to the dropbox for assessment. You are responsible for making sure your solutions are clear.
Q1 [10 marks] A bug of mass m crawls on a rotating turntable. The turntable rotates with a constant counter-clockwise angular speed of ω and the bug crawls in a counter-clockwise direction with a constant speed of v0 in a circle of radius R that is centered on the rotation axis of the turntable (see figure). There is a coefficient of friction µ between the bug and the turntable. Define the rotating reference frame such that the bug is located on the ˆı' axis at time t.
Figure 1: Left: Side view shows the 3-D coordinate system with the turntable in the non-inertial frame. The rotation axis is in the ˆk' direction. Right: Overhead view of the turntable with the bug’s path shown by the dashed circle. The ˆı' and ˆJ' coordinate vectors are shown at two different times to show the rotation of the non-inertial reference time with respect to an inertial observer (e.g., someone watching the bug on the turntable).
a) What is the vector equation for the bug’s radial and velocity vectors? Express these with respect to ˆı' and ˆj'.
b) What is the equation for the bug’s acceleration in the non-inertial frame? Explain your answer. Hint: consider how an observer on the turntable would describe the bug’s motion.
c) Which fictitious forces act on the bug? Explain how you know.
d) Find the magnitude and direction for all the fictitious forces acting on the bug.
e) Use your answers for parts b), c), and d) to find the equation for the net inertial force acting on the bug.
f) What is the maximum speed the bug can crawl without slipping? Hint: Only one force acts along the surface of the turntable.
Q2 [10 marks] A cannon fires a cannon ball of mass m in an eastward direction with an initial speed of v0 at an angle φ above the ground. The cannon is located at a latitude of θ in the Northern hemisphere. Using a system where ˆı' is East, ˆj' is North, and ˆk' is up, answer the following questions.
a) What are the equations for eastward and vertical velocity components as a function of time if we were to ignore the Earth as a rotating reference frame?
b) How long is the cannonball in the air? Assume the cannonball is fired from the ground and the surface below is flat.
c) Now consider the Earth as a rotating reference frame. What is the Coriolis force? Hint: Solve the vector cross product for the angular velocity vector and the velocity vector from a). Recall that ω = ω cos θˆj' + ω sin θˆk'. Describe the directions of the deflection.
d) If the cannon is located at the North pole (θ = 90◦ ), what is the direction of deflection from the Coriolis force?
e) For the cannon at the North Pole, what is the deflection for the total time in air and for what angle φ would you get the maximum deflection?