 [Homework Home > Rolling_and_Angular_Momentum] A solid sphere, having a mass M= kg and radius r= m is rolls with speed v= m/s toward a ramp. The moment of inertia of the sphere is I=\minifraction{2,5}Mr2 (where M and r are the mass and radius of the sphere). The ramp's angle is \theta= degrees . To what distance, h, will the ball rise up the ramp before coming to a stop? A block of mass m= kg is hanging from a rope attached to a pulley. The pulley has a mass M= kg and a radius of R= m . The block is held fixed, then suddenly released. What is the angular acceleration of the pulley? How long after being released will the block be falling at a speed of v= m/s ? (The moment of inertia of the pulley is I=\onehalf MR2.) A solid spherical ball, with moment of inertia I=\minifraction{2,5}MR2 rolls down the track as shown. It doesn't slip or bounce around at all, but rolls smoothly through the entire track (believe it or not!) M, the mass of the ball is kg and R, the radius of the ball is m . It starts at a height of H= m and leaves the track at a height of h= m . When it reaches the horizontal end part of the track, where the red star is, it flies off of the track. How far to the right of point A does the ball land? A figure skater spins with her arms outstretched with an angular speed of \omega= rev/s (left figure). The moment of inertia with her arms outstretched is I1= kg m2 . She then pulls their arms in, as shown in the right figure, decreasing her moment of inertia to I2= kg m2 . What is the skater's new rate of rotation? A boy with a mass of m= kg stands near the edge of a merry-go-round (MGR) which is not spinning. The system's (boy + MGR) total moment of inertia about the center is I= kg m2 . The boy standing at r= m from the center of the MGR suddenly jumps off in a tangential direction with a speed of v= m/s . How fast will the MGR be rotating when the boy leaves it? A boy runs directly toward the right as shown, and jumps onto a merry-go-round (MGR) which is initially at rest. He lands at the position of the blue dot, which is a distance d= m from the center of the MGR. The boy has a mass m= kg and runs with a speed of v0= m/s . The MGR has a mass of M= kg and a radius R= m . What angular speed does the MGR + boy have after the boy lands? Two disks are on a pole. The top one has moment of inertia I1= kg m2 and the bottom one has moment of inertia I2= kg m2 . Initially, only the bottom one is spinning at \omega0= rad/s2 . Suddenly, the top one is dropped onto the bottom one, and sticks to it without slipping. What is the final angular speed of the two-disk combination? A block of mass m= kg is attached to a rope. The rope goes through the center of a frictionless table. The block is moving with speed v= m/s along the dotted circle shown, that has a radius of r0= m . The rope is then pulled down as shown, until the radius that the block moves around is reduced to r1= m . What is the speed of the block around this smaller circle? A long uniform rod, of length L= m and mass M= kg can pivot freely about the red dot at its center, as shown. A bullet, with mass m= kg flies into the rod with speed v at an angle \theta= degrees as shown. If the bullet lodges into the rod and the rod is left spinning with \omega= rad/s immediately after the collision, how fast was the bullet moving? A solid red ball with radius rb= m and mass m= kg can roll along a track with a loop in it. The loop has a radius of r= meters . From what height h must the ball be rolled in order to make it around the loop? Assume there is no friction in the loop.