A rigid cross is made of two identical uniform thin rods, each of mass 1.2 kg and length 40 cm , joined at their centers at right angles. Four small point masses, each of mass 0.3 kg , are attached to the ends of the rods. The cross is mounted on a frictionless horizontal axle passing through its center. A light string is wrapped around a small cylindrical hub of radius 10 cm fixed to the center of the cross, and a b

Step-by-step solution

Moment of inertia of the two rods about the central axis: I_ rods = 2 ( 1 12 M_ r L^2 ) = 2 1 12 1.2 (0.4)^2 = 0.032 kg m ^2 . Moment of inertia of the four point masses at distance L/2 : I_ masses = 4 m ( L 2 )^2 = 4 0.3 (0.2)^2 = 0.048 kg m ^2 . Total moment of inertia of the cross, I = I_ rods + I_ masses = 0.032 + 0.048 = 0.08 kg m ^2 . Let T be the tension in the string and a be the downward acceleration of the block. Equation of motion for the block: M_ b g - T = M_ b a Equation of motion for the cross: TR_ h