Practice Exam 3 PHYS 221 PDF

Name: Practice Exam 3 PHYS 221 November 16th, 2024 You will have 55 minutes to complete this exam. Please leave the area to the right of the solid bla...

Name: Practice Exam 3 PHYS 221 November 16th, 2024 You will have 55 minutes to complete this exam. Please leave the area to the right of the solid black line clear: this area is reserved for grading. Read the full text of each question before answering it. Calculators are not allowed on this exam. Use the coordinate system given at the top right of each page. You must show your work to receive full credit. This includes starting from general equations and including units throughout numeric calculations. The following are provided on this exam: m g = 10 s2 1 ⃗r = ⃗r0 + ⃗v0 t + ⃗at2 ⃗v = ⃗v0 + ⃗at vf2 = v02 + 2 · ⃗a · ∆⃗r 2 ∆⃗r d⃗r ∆⃗v d⃗v v̄ = ⃗v = ā = ⃗a = ∆t dt ∆t dt P⃗ F = m⃗a F⃗ = − dU dr r̂ m1 m2 F⃗g = m⃗g F⃗G = −G 2 r̂ F⃗s = −k∆⃗x F⃗kf = −FN µsf v̂ Fsf ≤ FN µsf r dW W dW = F⃗ · d⃗r P = P̄ = dt ∆t P P p⃗0 = p⃗f p⃗ = m⃗v The following are NOT provided on this exam: 1 1 1 E0 + W0→f = Ef E = Kℓ + KR + U Kℓ = mv 2 KR = Iω 2 Ug = mgh Us = k∆x2 2 2 2 1 2 θ⃗ = θ⃗0 + ω ⃗ 0t + α ⃗t ω ⃗ =ω ⃗0 + α⃗t vf2 = v02 + 2 · α⃗ · ∆θ⃗ 2 ∆θ⃗ dθ⃗ ∆⃗ω d⃗ω ω̄ = ω ⃗ = ᾱ = α ⃗= ∆t dt ∆t dt 2 v 1 ω st = rθ vt = rω at = rα ac = t f= = r T 2π ⃗τ = ⃗r × F⃗ ⃗ = I⃗ω I∥ = I0 + mr2 P L 1 You do not need to show your work on Questions 1 and 2. 1. Three otherwise identical mass and spring systems are placed in different liquids such that system A is under damped, system B is critically damped, and system C is over damped. The three springs are compressed the same amount and the masses are released from rest at the same time. In which system or systems will the mass first pass the equilibrium point? 2. An object is rolling without slipping while moving in the positive î-direction. If the object is slowing down, in what direction is ω ⃗? 3. Consider the oscillation depicted in the graph below (you can annotate the graph to show your work): (a) What is the period of oscillation? (b) What is the amplitude of oscillation? (c) What is the maximum acceleration the oscillating object experiences? 2 4. Consider the image below, with four forces (arrows) acting on three beams rigidly connected to a central pivot point. For the forces, one grid unit represents one Newton. For the beams, one grid unit represents one meter. If the moment of inertia of the object is 12 kg m2 , what is α ⃗ for this system at this moment? 5. Consider an object with mass m, radius r, and moment of inertia I released from rest at a height h0 the top of a curved ramp. The object rolls on the ramp without slipping, eventually leaving the ramp travelling directly upwards and reaching a maximum height in the air of h2. (a) If h2 = 2h1 , what is h0 ? Answer in terms of m, r, I, g, and h2. 3 (b) Demonstrate that your answer to part (a) has the correct dimensions. (c) Evaluate your result from part (a) in the special case that I = 0. Explain conceptually what this means and why the special case result makes sense. 4 A pottery wheel (a solid disk with Iw = 21 mw rw2 ) is spinning (ω = ω0 ) in the horizontal plane. A large ball of clay Ic = 25 mc rc2 is dropped onto the wheel and sticks to the wheel, centered on a point halfway between the center and edge of the wheel. mc = 12 mw and rc = 21 rw. (a) What is the angular speed of the system after the clay ball lands on the wheel? Express your answer in terms of just ω0. (b) What fraction of the system’s (rotational) energy was lost? 5