A cylinder with rotational inertia I1 = 2.0 kg·m2 rotates clockwise about a vertical axis through its center with angular speed ω1 = 5.0 rad/s. A second cylinder with rotational inertia I2 = 1.0 kg·m2 rotates counterclockwise about the same axis with angular speed ω2 = 8.0 rad/s . If the cylinders couple so they have the same rotational axis what is the angular speed of the combination? What percentage of the original kinetic energy is lost to friction?

Aparticle's position is given by x = 3.00 - 9.00t + 3t2, in which x is in meters and t is in seconds. (a) what is its velocity at t = 1 s? (b) is it moving in the positive or negative direction of x just then? (c) what is its speed just then? (d) is the speed increasing or decreasing just then? (try answering the next two questions without further calculation.) (e) is there ever an instant when the velocity is zero? if so, give the time t; if not, answer "0". (f) is there a time after t = 3 s when the particle is moving in the negative direction of x? if so, give the time t; if not, answer "0".