# 1 equals the coefficient of kinetic friction between

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1) equals the coefficient of kinetic friction between a uniform chain and a horizontal tabletop. The chain has a length of L and is initially resting on the table. If a length of chain equal to 2 L / 1 is pulled off the table and given a slight tug downward, the chain will begin to slip off the table (as shown in the figure below) The chain sliding off the table such that the length of chain off the table (denoted by x as positive below the table) satisfies the differential equation x ̈ 1 g L x g for x 0 x L , where t is time and g 32 ft/sec 2 is the acceleration due to Earth’s constant gravity. Solve this equation for x (as a function of t , L , g , and ) given that the chain starts from rest at x 0 2 L / 1 and then determine the time T (in terms of , L and g ) when all the chain has left the table. ——————————————————————————————————————— 2
——————————————————————————————————————— Problem # 3 ( 25 Points ) - A Cantilever Beam A constant horizontal force F is applied to the free end (right end) of the uniform cantilever beam of length L and weight W , as shown in the figure below. The force F is horizontal, pointing to the left and applied at the right end at x L of the beam. The left end at x 0 is fixed to the wall. It can be shown that the vertical deflection y y x of the beam, as a function of x measured along with beam with x 0 as the left end and x L as the right end, satisfies the equation y ′′ x 2 y x

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