Cylindrical Roller

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Physics / How do you find maximum angular speed given the following?

In a manufacturing process, a large, cylindrical roller is used to flatten material fed beneath it. The diameter of the roller is 1.00 m, and, while being driven into rotation around a fixed axis, its angular position is expressed as

θ =2.50t^2 - 0.600t^3

where θ is in radians and t is in seconds. (a) Find the maximum angular speed of the roller. (b) What is the maximum tangential speed of a point on the rim of the roller? (c) At what time t should the driving force be removed from the roller so that the roller does not reverse its direction of rotation? (d) Through how many rotations has the roller turned between t=0 and the time found in part (c)?

(a) angular speed w = dθ/dt = 5t - 1.8t^2
dw/dt = 5 - 3.6t = 0 for max w
so max w occurs at t = 5/3.6 s = 1.39s
so w max = 5*1.39 - 1.8*(1.39)^2 = 3.47 rad/s

(b) tangential speed v = r*w
r = D/2 = 0.5m
so v = 0.5*w = 1.74 m/s

(c) w is positive until 5t = 1.8t^2
so t = 5/1.8s = 2.78s (or t = 0 invalid)
After t = 2.87s, w is negative (starts reversing direction of rotation)
Driving force would actually have to be removed some time before t=2.78s because the roller can't stop instantaneously, but insufficient info to calculate this.

(d) Up to t = 2.78s, θ = 2.5*(2.78)^2 - 0.6*(2.78)^3 rad = 33.95 rad = 5.40 rotations

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