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Solving Mechanics Problems Using Newton's Laws

Learn how to apply Newton's laws to solve mechanics problems efficiently and accurately in a physics setting. Explore concepts such as rotational motion, friction, and acceleration with practical examples and detailed explanations.

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Solving Mechanics Problems Using Newton's Laws

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  1. Example: A cylinder of mass M and radius R rolls down an incline of angle θwith the horizontal. If the cylinder rolls without slipping, what is its acceleration? N fS mg Newton's 2nd law for rotation: Rolling without slipping: Newton's 2nd law for translation of the CM: compare to g sinθ, the results for an object sliding without friction θ

  2. Example: A disk of radius R and mass M that mounted on a massless shaft of radius r << R and rolling down an inclinewith a groove. What is its acceleration? N fS R M m ~ 0 r Mg Very small if R >> r !

  3. Example: Loop-the-loop. A cart marble of radius r is released from height h in a roller coaster with a loop of radius R. What is the minimum h to keep the cart on the track? If no slipping, and r << R: new term (1) At the point B: A The minimum velocity is fixed by N = 0: B (2) h mg+N Combining Eq. 1 & 2: R (Without rotation the factor is 2.5)

  4. Example: 1 m of cord is wound around a flywheel of radius, 0.25 m and mass, 2 kg which is concentrated in the rim. The cord is pulled with a force of 100 N and drives the wheel without slipping. What is the final angular velocity of the wheel when the cord is exhausted? What is the period? d=1 m The pulling of the cord implies a fixed amount of work done on this system. If we equate this with the rotational energy of the flywheel, then we can discover how fast it is turning. F=100 N m=1 kg I=mR2 v=Rω W = K R=0.25m W = Fd

  5. Example: Two wheels with fixed hubs, each having a mass of 1 kg, start from rest, and forces are applied as shown. Assume the hubs and spokes are massless, so that the moment of inertia is I = mR2. In order to impart identical angular accelerations, how large must F2 be? 1. 0.25 N 2. 0.5 N 3. 1 N 4. 2 N 5. 4 N

  6. Example: Two identical disks have a string coiled around them. The string is pulled with a constant force F over a distance d. In one case, the disk rolls without slipping on a table. In the other case, the disk rotates about its axis (like a pulley). Compare the final angular speeds of the disks. Same work (i.e., added energy) in both cases: W = Fd In case 2, this energy becomes only rotational kinetic energy In case 1, this energy becomes both translational and rotational kinetic energy Case 2 Case 1 F F • 1 > 2 • 1 = 2 • 1 < 2

  7. Work done by a torque (for pure rotations) Instantaneous power: F θ2 Average power: Work from θ1 to θ2: If torque is constant, A force F acts on an object as it rotates from θ1 to θ2. Axis r θ1 F Important! These expressions are only true for pure rotations (with a fixed axis of rotation, or in the frame of reference where the axis is at rest). Otherwise, the path by the point where the force is applied is not a circle.

  8. M = 10 kg R = 1.0 m A uniform flywheel moving at 600 rpm comes to a stop after 1000 turns, mostly due to air resistance. What is the average torque produced by air? Example: Work by friction

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