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Figure P10.85

86.A plank with a mass M = 6.00 kg rides on top of two identical solid cylindrical rollers that have R = 5.00 cm and m = 2.00 kg (Fig. P10.86). The plank is pulled by a constant horizontal force F of magnitude 6.00 N applied to the end of the plank and perpendicular to the axes of the cylinders (which are parallel).  The cylinders roll without slipping on a flat surface.  There is also no slipping between the cylinders and the plank.  (a) Find the acceleration of the plank and of the rollers.  (b) What friction forces are acting?

Figure P10.86

87.A spool of wire rests on a horizontal surface as in Figure P10.87.  As the wire is pulled, the spool does not slip at the contact point P. On separate trials, each one of the forces F1, F2, F3, and F4 is applied to the spool.  For each one of these forces, determine the direction the spool will roll.  Note that the line of action of F2 passes through P.

Figure P10.87

88.Refer to Problem 87 and Figure P10.87.  The spool of wire has an inner radius r and an outer radius R.  The angle between the applied force and the horizontal can be varied.  Show that the critical angle for which the spool does not roll is given by

If the wire is held at this angle and the force increased, the spool will remain stationary until it slips along the floor.

89.In a demonstration known as the ballistics cart, a ball is projected vertically upward from a cart moving with constant velocity along the horizontal direction.  The ball lands in the catching cup of the cart because both the cart and ball have the same horizontal component of velocity. What If? Now consider a ballistics cart on an incline making an angle with the horizontal as in Figure P10.89.  The cart (including wheels) has a mass M and the moment of inertia of each of the two wheels is mR2/2.  (a) Using conservation of energy (assuming no friction between cart and axles), and assuming pure rolling motion

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