Columbia University

PHYS UN3003

Answer all questions in INK in the blue booklets provided. Be sure to PRINT your name neatly on the front of each booklet you use. Begin each question on a new page. Show ALL work to receive full credit. The test is out of a total of 100 points. Attempt all 5 questions. The points for each question are shown in brackets beside the questions. A formula sheet is atttached at the end of the test paper. NO calculators or other electronic devices are allowed to be used during the test. (20 pts) (1) A thin uniform stick of length L amd mass M lies at rest on a frictionless table. A particle of mass m moving with speed v0 perpendicular to the stick collides elastically with the stick, striking a distance d below the stick’s midpoint (see the view from above in the figure below). (a) Show that the moment of inertia for the stick to rotate about an axis through its center is given by I = 1/12 ML2 . (b) What quantities are conserved in the collision? Write down the conservation equation for each conserved quantity. (c) If the particle is to have zero velocity immediately after the collision, what is the value of mass m in terms of the other given variables? (10 pts) (2) Consider the “saw tooth” periodic function shown below. Assume that the magnitude of fmax is one. In that case, the function f(t) is specified by f(t) = t in the range from t = - 1 s to t = +1s, and repeating outside this range as shown.