Exam 2 March 12, 2014
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1 Exam 2 Instructions: You have 60 minutes to complete this exam. This is a closed-book, closed-notes exam. You are allowed to use a calculator during the exam. Usage of mobile phones and other electronic communication devices is NOT permitted during the exam. Problem 1 (20 pts) 2 (20 pts) 3 (20 pts) Total Score Name: Section (circle): 8:30-9:20 11:30-12:20 2:30-3:20 Exam 2 Page 1 of 9
2 Problem 1 (20 points): Given: The coefficient of static friction between the conical surface of a dish (cross section shown below) and a block of mass m is µ S = Find: (a) Sketch a free-body diagram of the block. (b) Determine the minimum constant angular velocity Ω required so that the block does not slide downwards. TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAA Exam 2 Page 2 of 9
3 This page is for extra work related to Problem 1. Exam 2 Page 3 of 9
4 Problem 2 (20 points): Given: Blocks A and B are connected by a rigid rod and are constrained to move within smooth slots, which are oriented vertically and horizontally, respectively. A spring of stiffness k is attached between block A and ground. At the instant shown below, the spring is unstretched, the blocks are released from rest and a force F is applied to block B in the horizontal direction. Block A has a mass of 2m, and block B has a mass of m. Find: (a) Express the velocity of block B written in terms of the velocity of block A. (b) Find the speed of block A when s A = 0. A B Exam 2 Page 4 of 9
5 This page is for extra work related to Problem 2. Exam 2 Page 5 of 9
6 Problem 3 (20 points): Part A: (10 points) Given: Particle P (of mass m p ) is released from rest and slides down a smooth, curved rod and sticks to block A (of mass m A ). Position 1 corresponds to the initial position of P; position 2 and 3 correspond to the position of P right before and after impact, respectively. Position 4 corresponds to the position of P and A when the spring is at maximum deflection. Answer the following questions related to the motion of P from positions 1 to 4. h Work done by the normal force between the particle and rod, U (N) 1 2 : (a) U N 1 2 > 0 (b) U N 1 2 = 0 (c) U N 1 2 < 0 (d) more information is needed about the shape of the guide in order to determine the sign of U N 1 2. Change in gravitational potential, V gr = (V 2 ) gr (V 1 ) gr : (a) V gr = m P gh (b) V gr = m P gh (c) V gr = m P gh (d) V gr = m P gh (e) V gr = 0 (f) more information is needed about the shape of the guide in order to determine the sign of V gr. Exam 2 Page 6 of 9
7 Speed of P at position 2, v p2 : (a) v p2 = vp1 2 2gh (b) v p2 = 2gh (c) v p2 = 2gh (d) v p2 = 2gh (e) v p2 = 2gh (f) v p2 = 0 (g) more information is needed about the the problem in order to determine v p2. Speed of P at position 3, v p3 : m A (a) v p3 = 2gh m p + m A (b) v p3 = m p m p + m A 2gh m p (c) v p3 = 2gh m p + m A (d) v p3 = 0 (e) more information is needed about the the problem in order to determine v p3. Maximum deflection of the spring at Position 4, : ma + m p m A (a) = 2gh k m A + m p ma + m p m p (b) = 2gh 2k m A + m p 2gh (c) = m p k(m A + m p ) (d) = m p m A + m p 2ghk (e) = m p 2gh k (f) = 2ghm p k (g) more information is needed about the problem in order to determine. Exam 2 Page 7 of 9
8 Part B: (10 points) Given: The AR Drone quadrotor is vertically taking off with a constant velocity of V t, as shown below. The propeller highlighted in the figure has a constant angular velocity of Ω. In addition, the entire vehicle has a yaw rate of Ψ, which itself is changing at a rate of Ψ (Ψ is measured about the global, fixed, Z axis). z V t P h r Ω o d y x ψ Z Q Y X L Find: It is desired to use the following equations to determine velocity and acceleration of end point P of the propeller: v P = v O + ( v P/O ) rel + ω r P/O a P = a O + ( a P/O ) rel + α r P/O + 2 ω ( v P/O ) rel + ω ( ω r P/O ) Provide symbolic expressions in their simplest form (using the XYZ frame) for the terms included in the above equations and noted on the next page. Assume that the xyz frame is attached to the quadrotor frame and NOT the propeller. (Note: you do NOT need to solve for v P and a P.) Exam 2 Page 8 of 9
9 (a) v O = (b) a O = (c) ( v P/O ) rel = (d) ( a P/O ) rel = (e) ω = (f) α = (g) r P/O = Exam 2 Page 9 of 9
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