Question

2012 AP PHYSICS C: MECHANICS FREE-RESPONSE QUESTIONS Sliding Only Sliding and Rotating Rotating with No Sliding No Friction Friction with Coefficient μ 1-0 Mech. 3. A ring of mass M, radius R, and rotational inertia MR is initially sliding on a frictionless surface at constant velocity uo to the right, as shown above. At time 10 it encounters a surface with coefficient of friction μ and begins sliding and rotating. After traveling a distance L the ring begins rolling without sliding. Express all answers to the following in terms of M, R-u,.μ. and fundamental constants, as appropriate. (a) Starting from Newton's sccond law in cither translational or rotational form, as appropriate, derive a differential equation that can be used to solve for the magnitude of the following as the ring is sliding and rotating i· The linear velocity u of the nng as a function of time t ii. The angular velocity w of the ring as a function of time (b) Derive an expression for the magnitude of the following as the ring is sliding and rotating i. The linear velocity v of the ring as a function of time i. The angular velocity a of the ring as a function of time time it takes the ring to travel the distance L (c) Derive an expression for the d) Derive an expression for the magnitude of the velkcity of the ring immediately afher it has traveled the distance ! (e) Derive an expression for the distance

Answer 1

(a) The normal force N is The net force is -f= Ma Heref uN uNMa Mg Ma a =-ug dv Since a - dt dv =-ug dt Here, fis the frictional force. The net torque for the rigid body is, T fR Also, r Ia fR Ia I fing uMgR MR2 do Since dt da dt R

(b) i. The final velocity is: at Here, a-g 11. The angular velocity is: g and Here, a R = 0 t o, =0 + ug (c) The velocity wlhen the ring travel the distance L is, V gt Use v R when slipping of the ring stops gt Here, o,R _ R HgR =v-g R ugt =vo-gt Vo2ugt t= 2ug

(d) The velocity of the ring immediately after it travelled distance L is: Here, t 2ug 2ug