Question

12. (15 points) A uniform, solid cylinder of mass M and radius R rotates on a frictionless horizontal axle (see figure below). Two objects with equal masses m hang from light cords wrapped around the cylinder. If the system is released from rest, find the following 12 i2 a) (6 points) The tension in each cord (Use any variable or symbol stated above along with the following as necessary: g.) Hints: Two masses move down with constant accretion, while the cylinder will rotate about the frictionless horizontal axis. Use à Ra. to relate two accelerations (6 points) the acceleration of each object after the objects have descended a distance h (Use the following as necessary: M m, R, h, and g) b) c) (3 points) What is the total kinetic energy of the system after the objects have descended a distance h

Answer 1

mg - T = ma (1) & 2T times R = I alpha (2) {I = MR/2^2} & a = R alpha (3) By (1) (2) & (3) 3 & (2) 2TR = MR^2/2 times alpha/R rightdoublearrow T = Ma/4 By (1) mg - Ma/4 = ma mg = (m/4 + m) a rightdoublearrow mg = (M + um)/4 a a = 4 mg/(M + um) So T = M/4 times m mg/m + 4m a) rightdoublearrow T = Mmg/(M + 4m) b) rightdoublearrow Acceleration of objects of mass m is a = 4 mg/(m + 4m) Accelerating of cylinder M is = 0 C) By of system = loss in So K. E_Total = 2 mgh