m A solid sphere of mass 3m and radius 2R has a rope tied around it. The free end is wrapped around a pulley with mass and radius R which in turn suspends a block of mass m'. (a) What is the moment of inertia for the sphere and for the pul- ley/disk? (b) Draw an extended free-body diagram for the sphere and then find the tension in the rope pulling the sphere in terms of the linear acce- leartion of the rope and mass m. (c) Draw a simple free-body diagram of the hanging mass and solve for the tension in the other part of the rope in terms of the linear acceler- ation and the mass m'. (d) Now, draw an extended free-body diagram of the pulley and solve for the linear acceleration of the rope. Your answer should now only depend on the given masses. (e) Does your answer make sense if the pulley and sphere have no mass?

B6

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A long, thin, wire of uniform cross-section has Young's Modulus Y and mass density D. When laid horizontally on a bench its unstretched length is L. One end is attached to a fixed point and the rest of the wire is allowed to hang freely and vertically in the earth's gravitational field. Find the extension of the lowest end of the wire (i.e., the wire's length additional to L) due to gravitational force. The acceleration due to gravity is g.

A\a = 60" B= ??/ B Figure (d): (D) is a homogeneous disk of mass 300 g and of center o. Calculate the tension in the strings AE and BC and the value of ß. We can use an axis of rotation passing through C. E (D)

A uniform aluminum beam 9.00 m long, weighing 300 N,rests symmetrically on two supports 5.00 m apart (Fig. ). A boyweighing 600 N starts at point A and walks toward the right. (a) In thesame diagram construct two graphs showing the upward forces FA andFB exerted on the beam at points A and B, as functions of the coordinatex of the boy. Let 1 cm = 100 N vertically and 1 cm = 1.00 m horizontally.(b) From your diagram, how far beyond point B can the boy walkbefore the beam tips? (c) How far from the right end of the beam shouldsupport B be placed so that the boy can walk just to the end of the beamwithout causing it to tip?

- 2a 2a o' 2a -2a Figure P4.17 4.17. Find the product inertia Iyy for the beam section shown in Figure P4.17. Notice that the centroid must be at O by symmetry.

finished in an analytical way

5. Car doors are easy to slam shut but difficult to press shut by hand force. The door lock as two latches, the first easily engaged, and the second requiring the rubber seal around the door to be quite compressed before it can engage. The rubber seal gives a spring constant of 50,000N/m at the lock position for the door, and the mass moment of inertia for the door around its hinges is 2.5kg- m². The distance between the lock and the hinges is Im. Calculate what force is needed to press the car door shut at the lock if a speed of 0.8m/s at the lock slams it shut.

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