Any help is appreciated! PLease show all work so I can understand the process, n

Question

Any help is appreciated! PLease show all work so I can understand the process, not just the answers. THANK YOU!!!! :)

1. A disk, with a radius of 0.25 m, is to be rotated like a merry-go-round through
800 rad, starting from rest, gaining angular speed at the constant rate ?1 through the first
400 rad and then losing angular speed at the constant rate - ?1 until it is again at rest. The
magnitude of the centripetal acceleration of any portion of the disk is not to exceed 400
m/s^2.
(a) What is the least time required for the rotation?
(b) What is the corresponding value of ?1?

Any help is appreciated! PLease show all work so I can understand the process, not just the answers. THANK YOU!!!! :) 1. A disk, with a radius of 0.25 m, is to be rotated like a merry-go-round through 800 rad, starting from rest, gaining angular speed at the constant rate ?1 through the first 400 rad and then losing angular speed at the constant rate - ?1 until it is again at rest. The magnitude of the centripetal acceleration of any portion of the disk is not to exceed 400 m/s^2. (a) What is the least time required for the rotation? (b) What is the corresponding value of ?1? 2. The angular acceleration of a wheel is alpha in radians per second-squared and t in seconds. At time t = 0, the wheel has an angular velocity of +2 rad/s and an angular position of +1 rad. Write expressions for: (a) The angular velocity (rad/s) (b) The angular position (rad) as functions of time (s). alpha= 6.0t^4 ? 4.0t^2, with

Explanation / Answer

a_rmax= 400 = r*w^2

wmax^2 = 400/0.25

w max= 40 rad/s

during the first 400 rad

theta = average speed*t1

400 = (40+0)/2*t1

t1 = 20 s

theta 2 = 400 = (40+0)/2*t2

t2 20 s

a) T = t1 + t2 = 80s

b) alfa = (w-0)/t1 = (0-w max)/ = 40/20 = 2 rad/s^2

===============

a) w = integration of 6t^4 - 4t^2

w = 6t^5/5 - 4t^3/3 + wo

at t=o w = 2rad/s

so

2 = 0 + wo=====> wo = 2

w = 6t^5/5 - 4t^3/3 + 2 <-----------answer

b) angular position = integration of w

angular position = integration of 6t^5/5 - 4t^3/3 + 2

angular position = theta = t^6/5 - t^4/3 + 2*t + thetao

at t = o.... theta = 1rad

so

1 = 0 + wo=====> thetao = 1

theta = = t^6/5 - t^4/3 + 2*t + 1

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