8. (a) A yo-yo is initially at rest on a horizontal surface. The string is then pulled vertically upwards by a force of magnitude F as shown in Figure 5, tending to make the yo-yo rotate clockwise. This causes the frictional force fbetween the yo-yo and surface to act horizontally to the right, thus making the yo-yo roll to the right without slipping. Figure 5 The yo-yo has a mass M, moment of inertia Iabout the axis of symmetry, inner radius r where the string is wound, and outer radius R ( Assuming that the yo-yo rolls without slipping, show that the magnitude of the frictional force,f is given by f-BF , where Î² is a constant you need to determine Hint: Consider the equations of motion of the yo-yo: for example, for the center of (ii) If the coefficient of static friction is Î¼, determine the maximum value of F so that explicitly in terms of r, R, Mand I. mass and rotation about the center of mass.] no slipping takes place. to gravity) and the constant Î² from part (i). Express your result in terms of Î¼, M, g (acceleration due (b) A ball, starting from rest, lls down a hill and then flies through the air at the end of the hill at point A as shown in Figure 6, finally landing back on the horizontal ground. 1.8 m 35 Figure 6 Treating the ball as a thin-walled spherical shell, determine the horizontal range x indicated Assume that the ball does not slip when i is rolling and ignore air