In order for an object to roll smoothly (with constant angular acceleration) down a ramp, it must have either spherical or cylindrical symmetry. Consider spherically and cylindrically symmetric objects with mass M and outer radius R rolling without slipping down an incline of height h and angle 0 from the horizontal. The rotational inertias of such objects can be written in the generalized form: / = CMR² , where c is a shape factor whose value depends on the geometry of the object. (If you look at the table of rotational inertias in the textbook you'll be able to see that this is true.) (For this question, consider M, R, c, h, 0, and g as the given quantities -express your answers in terms of some or all of these quantities. Simplify your answers as much as you can.) M a. If the object starts from rest at the very top of the ramp before rolling freely down the ramp without slipping, find the object's (linear) rotational speed at the bottom of the ramp. (Hint: use conservation of energy.)

In order for an object to roll smoothly (with constant angular acceleration) down a ramp, it must have either spherical or cylindrical symmetry. Consider spherically and cylindrically symmetric objects with mass M and outer radius R rolling without slipping down an incline of height h and angle 0 from the horizontal. The rotational inertias of such objects can be written in the generalized form: / = CMR² , where c is a shape factor whose value depends on the geometry of the object. (If you look at the table of rotational inertias in the textbook you'll be able to see that this is true.) (For this question, consider M, R, c, h, 0, and g as the given quantities -express your answers in terms of some or all of these quantities. Simplify your answers as much as you can.) M a. If the object starts from rest at the very top of the ramp before rolling freely down the ramp without slipping, find the object's (linear) rotational speed at the bottom of the ramp. (Hint: use conservation of energy.)