EMA 202, Fall 2014 Name: ______________________________ Midterm #1, closed book/notes, 90 min. Instructions • Write your name on every sheet. • Show all of your work. • To maximize opportunities for partial credit, solve problems symbolically to the extent possible and substitute numerical values at the end. • Include free body diagrams for all equilibrium equations. • This is a closed book examination. • The only notes allowed are the equations provided with this exam. • Calculators are allowed. • The use of cell phones is prohibited. • The instructors and the University of Wisconsin expect the highest standards of honesty and integrity in the academic performance of its students. It is important that the work submitted on this examination is yours and yours alone. • Receiving or giving aid in an examination or using notes on a closed note exam will be considered cheating and will result in a grade of F and the case being reported to the Dean of Students Office. Circle Your Discussion Section: 301 (Tu 1:20, David) 302 (Tu 2:25, David) 303 (W 3:30, David) 304 (Th 8:50, Matt) 305 (Th 12:05, Matt) 306 (Th 1:20, Matt) Grading: Q1 /25 Q2 /25 Q3 /25 Q4 /25 Total /100EMA 202, Fall 2014 Name: ______________________________ Midterm #1, closed book/notes, 90 min. Question 1 (25 points) Conceptual questions: please include a few sentences or equations to justify your answers 1A (6 points) The trajectory of a particle is shown on the figure below. At point A the speed of the particle is decreasing. Draw the acceleration and velocity vectors at point A (magnitude is not important) 1B (5 points) A windmill sits in an inertial coordinate system. It’s blades are rotating at a constant rate ω. Is the acceleration of point A equal to 0? A tuˆnuˆX y AEMA 202, Fall 2014 Name: ______________________________ Midterm #1, closed book/notes, 90 min. 1C (8 points) Determine the equation for the trajectory of a projectile, y(x). Assume x(t=0) = 0 and y(t=0) = 0. (hint: your equation should look like Y = C1X+C2X2 , Find C1 and C2) 1D (6 points) A toy car is riding a track with a circular loop of radius R. What is the minimum speed that the car must be going at point A to maintain contact with the track? Let m=car mass A gEMA 202, Fall 2014 Name: ______________________________ Midterm #1, closed book/notes, 90 min. Question 2 (25 points) A ball is thrown horizontally from the top of a 20m tall building with a velocity of 10m/s. A short time later, a person on the ground (point B) throws another ball such that the two balls collide at point C. Given that the person on the ground throws from a height of 1m and an angle θ = 35o, a) Find the height at which the balls collide b) Find the velocity of the ball thrown from point B c) Find the time delay between when ball A and B are thrownEMA 202, Fall 2014 Name: ______________________________ Midterm #1, closed book/notes, 90 min. (Additional workspace for Question 2)EMA 202, Fall 2014 Name: ______________________________ Midterm #1, closed book/notes, 90 min. Question 3 (25 points) A 10kg mass, m, rotates around a vertical pole in a horizontal circular path with radius R=1m. The angles of the ropes with respect to the vertical direction are 35o and 55o. Find the range of the velocity of the mass such that the mass will remain on the circular path described above? 35oEMA 202, Fall 2014 Name: ______________________________ Midterm #1, closed book/notes, 90 min. (Additional workspace for Question 3)EMA 202, Fall 2014 Name: ______________________________ Midterm #1, closed book/notes, 90 min. Question 4 (25 points) Crates A and B of mass 50 kg and 75 kg, respectively, are released from rest. The linear elastic spring has stiffness k=500Nm. Neglect the mass of the pulleys and cables and neglect friction in the pulley bearings. If µk=0.25 and the spring is initially unstretched, determine the speed of B after A slides 4 m.EMA 202, Fall 2014 Name: ______________________________ Midterm #1, closed book/notes, 90 min. (Additional workspace for Question 4)EMA 202, Fall 2014 Name: ______________________________ Midterm #1, closed book/notes, 90 min. Geometry: cCbBaAsinsinsin cBABAC cos2222 1sin cos22 Particle Rectilinear Motion: dtdva  dtdsv  vdvads For the special case of constant acceleration (ac), and assuming initial conditions are specified at t = 0: tavtvco)( 221)( tatvstscoo )(222ocossavv  Particle Curvilinear Motion: Cartesian Results: kjivˆˆˆzyx kjiaˆˆˆzyx Normal/Tangential Results: tsuvˆ ntvv uuaˆˆ2 where |/|])/(1[222/32dxyddxdy Polar Results: u uvˆˆ rrr u uaˆ)2(ˆ )(2rrrrr Spherical Results: uu uvˆ sinˆˆ rrrr uu uaˆ)cos2sin2sin( ˆ)cossin2(ˆ )sin(2222rrrrrrrrrr General Time Derivatives of a Vector: uωuˆ)(ˆut AωuAAAAtˆ)( )(ˆ2ˆ)( AωωAωuωuAAAAAAAAAt Principle of Work and Energy: U12 = T2 – T1 where U12 = r F2 1 d and T = 221mv Conservation of Energy (assumes only conservative forces): T1 + V1 = T2 + V2 (For rigid bodies, these results are unchanged except for the fact that T = 222121GGImv ) c b a C B AEMA 202, Fall 2014 Name: ______________________________ Midterm #1, closed book/notes, 90 min. V is potential energy and takes different forms depending on the source. For terrestrial gravity, V = mgy, (mgyG for a rigid body) while for a linear elastic spring, V = 221k Power: vFP Efficiency: inputpower outputpower  Linear Impulse and Momentum: 1221ppI where dt2 1 21FI and vpm (For rigid bodies, the linear impulse and momentum expressions are identical but with Gvv) Coefficient of restitution:       1122BAABvvvve Principle of Angular Impulse and