A cord connected at one end to a block which can slide on an inclined plane has its other end wrapped around a cylinder resting in a depression at the top of the plane as shown in Fig. 10–81. Determine the speed of the block after it has traveled 1.80 m along the plane, starting from rest.

(b) the coefficient of friction between all surfaces is μ = 0.035 . [Hint: In part (b) first determine the normal force on the cylinder, and make any reasonable assumptions needed.]

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Verified step by step guidance

1. Identify the forces acting on the block and the cylinder. For the block, consider the gravitational force parallel to the incline, the normal force perpendicular to the incline, and the frictional force opposing the motion. For the cylinder, consider its weight, the normal force from the depression, and the frictional force at the contact point with the cord.
2. Apply Newton's second law to the block and the cylinder separately. For the block, set up the equation considering the net force along the incline (taking into account gravitational force, frictional force, and tension in the cord). For the cylinder, consider the rotational dynamics and the frictional force that acts to prevent it from slipping.
3. Calculate the normal force on the cylinder using the assumption that the cylinder does not slip and the tension in the cord is uniform. Use this to find the frictional force on the cylinder using the given coefficient of friction.
4. Solve the equations simultaneously to find the acceleration of the block. This involves substituting the expressions for tension and frictional forces from the cylinder's equation into the block's equation.
5. Use the kinematic equation \(v^2 = u^2 + 2as\) to find the final speed of the block, where \(v\) is the final speed, \(u\) is the initial speed (zero in this case), \(a\) is the acceleration found from the previous step, and \(s\) is the distance traveled along the plane.