Problem 3 A hoop of mass m and radius R, rolls without slipping down an inclined plane of angle a. The inclined plane of mass M moves horizontally. Let X be the abscissa of point O' with respect to 0, and s be the abscissa of the center of mass of the rolling hoop with respect to O'. Given: the moment of inertia of the hoop is I = mR*, and the condition of rolling without slipping is expressed as s = R®: 1. Write the Lagrangian of the system. 2. Assuming that the Lagrangian of the system (inclined plane and hoop) is given by: = ms? +(m + M)X" + ms'X'cos(a) + mg(s)sin(a) 3. Write the expression of the Lagrangian function 4. Derive the Euler Lagrange equations 5. Find s" and X" in terms of the masses (m,M), angle a and g Referencelevel m,R M ex 2.

Problem 3 A hoop of mass m and radius R, rolls without slipping down an inclined plane of angle a. The inclined plane of mass M moves horizontally. Let X be the abscissa of point O' with respect to 0, and s be the abscissa of the center of mass of the rolling hoop with respect to O'. Given: the moment of inertia of the hoop is I = mR*, and the condition of rolling without slipping is expressed as s = R®: 1. Write the Lagrangian of the system. 2. Assuming that the Lagrangian of the system (inclined plane and hoop) is given by: = ms? +(m + M)X" + ms'X'cos(a) + mg(s)sin(a) 3. Write the expression of the Lagrangian function 4. Derive the Euler Lagrange equations 5. Find s" and X" in terms of the masses (m,M), angle a and g Referencelevel m,R M ex 2.