When a certain carousel is turned on, it rotates with a constant angular acceler

Question

When a certain carousel is turned on, it rotates with a constant angular acceleration of 0.700 rad/s2 for an unknown time interval. Then, it suddenly loses power, and decelerates at 0.500 rad/s2 until it has come to rest. During the entire process, a rider on a wooden horse 6.00 m from the center of the carousel makes 3.00 complete circles.

(a) How many revolutions has the rider made when the carousel loses power?

(b) What is the total time for the process?

(c) At the moment the rider has completed 1.00 rev, what are her radial and tangential accelerations?

Explanation / Answer

Here ,

a1 = 0.70 rad/s^2

a2 = - 0.50 rad/s^2

R = 6 m

theta = 3 * 2pi rad

a) let the maximum angular speed is w

for the total angle rotated

w^2/(2 * 0.70) + w^2/(2 * 0.50) = 3 * 2pi

solving for w

w = 3.32 rad/s

after losing power

angle rotated = w^2/(2 * a2)

angle rotated = 3.32^2/(2 * 0.50)

angle rotated = 11.02 radian = 1.75 revs

the angle rotated is 1.75 revolutions

b)

total time taken = w/a1 + w/a2

total time taken = 3.32/.70 + 3.32/.50

total time taken = 11.4 s

the time taken is 11.4 s

c)

after 1 rev

wf = sqrt(2 * 0.70 * 2pi) = 2.97 rad/s

Now, for radial and tangential acceleration

radial acceleration = w^2 * r

radial acceleration = 2.97^2 * 6

radial acceleration = 52.7 m/s^2

tangetial acceleration = a *r

tangetial acceleration = 0.70 * 6

tangetial acceleration = 4.2 m/s^2

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