Large Hoop
How far does the roll hoop?
In a circus performance, a large 4.4kg ring with a radius of 2.4 m rolls without slipping. If the ring is given an angular velocity of 6.0 rad / s while rolling on the horizontal axis and is allowed to roll down a ramp inclined 10 ° to the horizontal extent (measured along the slope) makes the roll hoop?
Sea: m the mass of the ring, r the radius of the ring, I will be the moment of inertia of the ring around an axis through its center, g acceleration of gravity, s the distance along the ramp, not the slope of the ramp to the horizontal, w be the angular velocity of the ring, the velocity v linear hoop. Equating the initial kinetic energy of translation and rotation in the final potential energy (2.1) (mv ^ 2 + iw ^ 2) = mg sin (t) … (1) the linear velocity on the angular velocity: v = rw … (2) The moment of inertia of the ring is: i = mr ^ 2 … (3) Substituting in (1) for v in (2) and the i of (3): mgs sin (t) = (1.2) (mr ^ 2 w ^ 2 + mr ^ 2w ^ 2) = 2 ^ w ^ mr ^ 2 rs = 2 w ^ 2 / [sin g (t)] = 2.4 ^ 2 * 6.0 ^ 2 / [9.81 sin (10)] = 122 m.
