1 / 8

f S

Example: A cylinder of mass M and radius R rolls down an incline of angle θ with the horizontal. If the cylinder rolls without slipping, what is its acceleration?. N. f S. mg. Newton's 2 nd law for rotation:. Rolling without slipping:. Newton's 2nd law for translation of the CM:.

tdora
Download Presentation

f S

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

More Related