1. (3 pts) Moment of Inertia. Consider a uniform thin rod of mass M and length & but pivoted about an axis which can be anywhere along the rod's length. Call the distance from the left end of the rod to the axis about which the rod pivots, d. a. (2 pts) By integrating over the entire rod, what is the moment of inertia about the axis? b. (1 pts) What is the minimum value the moment of inertia can have? Prove your answer by explicitly differentiating your answer to part a. What is the value of d at this point? 2. (4 pts) Friction and Incline. Two blocks lic on a horizontal M, N and a tilted surface and are attached to each other by a massless string as shown in the figure to the right. Block m, lies on a surface with a coefficient of kinetic friction a, while the other block lies on a frictionless surface. The string rolls over a pulley (a uniform disk of mass M and radius R) and does not slip, causing the pulley to rotate as the blocks move. a. (1 pt) Draw the free-body diagrams for all three bodies. Include in cach diagram the coordinate system you will use for part b. b. (1 pt) Write Newton's Second Law along cach coordinate that you drew in part a. c. (1 pt) What is the acceleration of block m, down the incline? d. (1 pt) What is the coefficient of kinetic friction if the blocks move at constant speed? 3. (5 pts) Rolling from the Top. A spherical ball of mass M and outer radius R is pulled by a force Fapplied to the very top of the ball. Unlike most balls you'll deal with in this class, this ball does not have uniform mass density, but rather has a moment of incrtia / = AMR , where / is obviously a dimensionless constant. The surface has a coefficient of static friction, /,, which is sufficiently large that the ball never slips against it. a. (1 pt) Draw the free-body diagram of the ball, including all necessary coordinates. b. (1 pt) Write Newton's Second Law along cach coordinate you drew in part a. c. (1 pt) What is the acceleration of the ball in terms of F, M, R, a,, /, and g, or a subset thereof? d. (1 pt) What is the largest force F with which you can pull without the ball slipping, again as a function of M, R, A,, /, and g, or a subset thereof? 2. (1 pt) If you have solved this problem correctly, you will find that the direction of the frictional force depends upon the moment of inertia. What is the critical value of / at which the frictional force required for the ball not to slip goes to zero? (That is, where is the boundary between static friction pointing to the left, and pointing to the right?) Describe a mass distribution that would give you this value of /.4. (3 pts) Ball Hits Stick. A marble of mass m traveling at speed v. strikes the end of a stationary stick of mass Af and length & which was initially pointed perpendicular to the marble's path. After the collision, the marble stops dead in its tracks, while the rod moves away both translating and rotating. a. (1 pts) After the collision, what is the velocity of the stick's center of mass? b. (2 pts) After the collision, what is the stick's angular velocity about its center of mass? 5. (6 pts) Colliding Disks. A uniform disk of mass m, and radius R, is traveling initially with speed v. directly to the right when it just barely touches the edge of a second uniform disk of mass m, and radius R which was initially stationary. The two disks then remain stuck to each other as shown, while translating to the right and rotating. In questions below, "system" means both disks. a. (1 pt) After the collision, how far is the system's center of mass from the center of m,? b. (1 pt) Before the collision, what was the angular momentum about the system's center of mass? c. (1 pt) After the collision, what is the system's moment of inertia about the system's center of mass? d. (1 pt) After the collision, what is the velocity of the system's center of mass? 2. (1 pt) After the collision, what is the angular velocity about the system's center of mass? f. (1 pt) How much kinetic energy was lost during this collision? 6. (6 pts) Game of Throws. A rod of mass M and length L covered in Velcro@ loops hangs from a frictionless pivot and is initially motionless. You then throw a ball of mass in covered in Velcro@ hooks so that it sticks to the rod. Your goal is to throw the ball fast enough so that the Velcro system of ball plus rod goes all the way around a complete loop. a. (2 pts) First, you throw the ball so that it hits the end of the rod, as Incoming shown in the figure. What is the minimum speed with which you ball must throw the ball in order for the rod to make it over the top so that it can go all the way around? b. (4 pts) Certainly hitting the rod at the end, as opposed to the middle, imparts more angular momentum to the rod. However, sticking the ball at the end, as opposed to the middle, increases the energy required to raise the center of mass of the system when it goes over the top. For balls comparable in mass to the rod, hitting the rod at a smaller distance d from the axis (d/L