Please help? In a manufacturing process, a large, cylindrical roller is used to

Question

Please help?

In a manufacturing process, a large, cylindrical roller is used to flatten material fed beneath t. The diameter of the roller is 3.00 m, and, while being driven into rotation around e fixed axis, its angular position is expressed as theta = 2.70t^2 - 0.750t^3 where theta is in radian and t is in seconds. (a) Find the maximum angular speed of the roller. rad/s (b) What is the maximum tangential speed of a point on the of the roller? m/s (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)?

Explanation / Answer

(a) angular speed w = d/dt = 5.4t - 2.25t^2
dw/dt = 5.4 - 4.5t = 0 for max w
so max w occurs at t = 5.4/4.5 s = 1.2s
so w max = 5.4*1.2 -2.25*(1.2)^2 = 6.48-3.24=3.24 rad/s

(b) tangential speed v = r*w
r = D/2 = 0.5m
so v = 0.5*w = 1.62 m/s

(c) w is positive until 5.4t = 2.25t^2
so t = 5.4/2.25 = 2.4s (or t = 0 invalid)
After t = 2.4s, w is negative (starts reversing direction of rotation)
Driving force would actually have to be removed some time before t=2.4s because the roller can't stop instantaneously, but insufficient info to calculate this.

(d) Up to t = 2.4s, = 2.7*(2.4)^2 - 0.75*(2.4)^3 rad =15.552+10.368= 25.92 rad=25.95/2*3.14= = 4.12 rotations

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