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Final: 2 hour long. Final 221 Rough estimate of a 120-125 points allocation: 1. Problem from Flipit homework: 35-45 pts 2. Lecture slides: Proofs/problems: 60-70 pts 3. Cumulative section: questions from previous exams (mostly): 15-20 pts Final 221 Ch14 Rotational Kinematics (Lecture slides): β’ Define a Rigid body (4pts) β’ Proof: Prove that for pure rotation around a fixed axis (8pts) K= 12 I/o w/o2 where I/o =Smiri2 β’ Problem: Compute the moment of inertia of a rod around a perpendicular axis through its end (12pts) Ch15 Parallel Axis theorem and Torque (Lecture slides): 1 1 2 2 β’ Proof: Starting πΎπππ‘ = 2 ππ£πΆπ + 2 ππ=1 ππ π£π/πΆπ (a Ch13 result or see βOptional 1 1 2 proofβ in CH15 slides) prove that πΎπππ‘ = 2 ππ£πΆπ + 2 πΌπΆπ π2 for a rigid body. Use this result to prove the Parallel Axis Theorem. (14pts) β’ Know how to compute the torque from Ο=r Ξ§ πΉ β’ Optional proof: proof mentioned above (bonus points) Ch16 Rotational Dynamics: No required proofs, only problems (Lecture slides): β’ Problem: Atwood machine problem (disk with strings and 2 masses hanging) (16pts) β’ Problem: Disk rolling down incline: know the 3 different approaches to the solution: point of contact solution, CM solution and energy solution. (14-18pts) β’ Optional proof: proof that the CM torque equation is valid even when the CM is accelerated. (bonus points) Final 221 Ch17/18 Rotational Statics (Lecture slides): β’ If a rigid body is static then Ο=0 and πΉ = 0 around point of choice β’ Problem: hanging sign (14 pts) β’ Problem: Ladder against wall (16 pts) β’ Problem: Box in truck problem (16 pts) Final Review Ch 19/20 (Lecture slides): dL β’ Proof: Prove that dt ο½ ο΄ starting from L ο½ r ο΄ p and then prove that angular momentum is conserved when no external torque is present. (8pts) β’ Problem: Use πΏ = πΆππ to solve the following problems (you must justify and demonstrate that the net torque is zero to get any credit): β’ person jumping on a merry-go-round (8pts) β’ skaters (4pts) β’ orbits (8 pts) β’ the mass and string through table problem (14 pts) β’ Problem: Derive the precession angular frequency for a torque constantly perpendicular to L (14 pts) Final Review Ch21 Simple Harmonic motion (Lecture slides): β’ Problem: Starting from F=ma derive the differential equation satisfied by a mass m, attached at the end of a horizontal spring of constant k. Show that: π₯ π‘ = π₯0 cos ππ‘ + π is a solution provided that π takes a special value. Derive that value in terms of π and π. (12 pts) β’ Problem: Vertical spring of constant k with hanging mass m. Derive, from F=ma the differential equation satisfied by ΞΎ=x-xequil. and find solution, and period in terms of m and k. β’ Using a second order Taylor series expansion for the potential energy π’ π₯ associated with a force F in one dimension, prove that any system will behave like a spring when disturbed from a stable equilibrium position. Derive the value of the Spring constant k in terms of: π’β²β² π , the second derivative of the potential evaluated at the equilibrium position x=a. (12pts) Final Review Ch 22 Physical Pendulum (Lecture Slides): β’ Problem: Derive, from the torque equation, the solution for π t and the angular frequency of a torsion pendulum subject to a torsion torque π = πΎπ where K is a constant. (16 pts) β’ Problem: derive from the torque equation, the equation satisfied by π, the angle from the vertical of a physical pendulum of inertia πΌ0 . Find the solution for ΞΈ for small oscillations, explaining why the approximation is needed. Then derive the angular frequency of the pendulum in terms of I and other physical parameters (16pts) Ch 23 Harmonic Waves and Wave Equation (Lecture Slides): β’ Be able to justify if a function f(x,t), of time and space, depicts a wave travelling down the x axis. Know how to determine the velocity of the wave given the function. (8pts) β’ Proof: Prove that if a wave is periodic in time with period T, it must also be periodic in space with wavelength π. In doing so derive the relationship between π and π (10 pts) β’ Problem: Derive, starting from F=ma, the wave equation describing a small kink travelling down a string of mass density π under tension T Show that the equation admits solutions of the form π¦ π₯, π‘ = π΄ πππ π π₯ β π£π‘ provided π£ = π π . (20 pts) Final Review Ch 24 Superposition of waves (Lecture Slides): β’ Proof: Show that two harmonic waves of the same amplitude travelling in opposite direction(say π = π΄ cos π π₯ ± π£π‘ ), can generate standing waves. Derive the position of the nodes in terms of k. (16 pts) Ch 25 Static fluids (Lecture slides): πΉ β’ Proof: By considering a static fluid element prove that dπ = ππdπ¦ where P = π΄π and π¦ is the vertical position (12 pts) β’ Problem: Derive the max height that a pump can lift a fluid of density π at the surface of the earth (12 pts) Ch 26 Moving Fluids (Lecture Slides): β’ Proof: Use the work energy theorem to derive Bernoulliβs equation (20 pts) β’ Problems (each 6 pts; use Bernoulli to explain the phenomenon): β’ Airplane wings β’ Roofs in high wind β’ Lift/drop on tennis balls