Question
In figure 10-46, the incline is frictionless and the stringpasses through the center of mass of each block. The pulley has amoment of inertia I and radius R. A.) Find the net torque acting on the system (the two masses,string, and pulley) about the center of the pulley. B.) Write an expression for the total angular momentum of thesystem about the the center of the pulley. Assume the masses aremoving with a speed v. C.) Find the acceleration of the masses by using your resultsfor Parts(a) and (b) and by setting the net torque equal to therate of change of the system's angular momentum. The picture is a basic wedge with m1 hanging off connected bya string going up and over a pulley at the top of the inclineconnected to m2 on the inclination of the wedge. In figure 10-46, the incline is frictionless and the stringpasses through the center of mass of each block. The pulley has amoment of inertia I and radius R. A.) Find the net torque acting on the system (the two masses,string, and pulley) about the center of the pulley. B.) Write an expression for the total angular momentum of thesystem about the the center of the pulley. Assume the masses aremoving with a speed v. C.) Find the acceleration of the masses by using your resultsfor Parts(a) and (b) and by setting the net torque equal to therate of change of the system's angular momentum. The picture is a basic wedge with m1 hanging off connected bya string going up and over a pulley at the top of the inclineconnected to m2 on the inclination of the wedge.
Explanation / Answer
Let the system include the pulley,string,and the blocks &assume that the mass of the string is negligible.The angularmomentum of this system changes because a net torque acts onit. a) let us express the net torque about the center of massof the pulley as net = Rm2gsin -Rm1g =Rg(m2sin - m1) b) The total angular momentum of the system about an axis throughthe center of the pulley is L =I+m1vR+m2vR = vR(I/R2 + m1 + m2) c) let us express as the time derivative of theangular momentum as =dL/dt = d/dt[vR(I/R2+m1+m2)] =aR(I/R2+m1+m2) equating the results of part(a) and solving for a weobtain a =g(m2sin-m1)/I/R2+m1+m2