Eight students, each with a mass m are standing on the perimeter of a rotating p

Question

Eight students, each with a mass m are standing on the perimeter of a rotating playground merry-go-round of mass of 4m and radius R. Simultaneously they all step in towards the center to a point R/3 from the axis of rotation. Treating the students as point masses and the merry-go-round as a solid cylinder, find:

a. the ratio of the final to initial angular speeds: Z f Zi ,

b. the ratio of the final to initial centrifugal force (i.e. the force needed to hang on) felt by each student: FCf FCi ,

c. the final speed in radians per second if the original speed was 1/4 rev/s, and

d. the initial and final centrifugal forces for m = 65 kg and R = 2.0 m. Will they be able to hang on?

Explanation / Answer

Here ,

for the initial moment of inertia , Ii = 0.5 * 4m * R^2 + 8 m * R^2

Ii = 10 * m * R^2

If = 0.5 * 4m * R^2 + 8 * m * (R/3)^2

If = 26/9 * m * R^2

a) Using conservation of momentum

Ii * Zi = If * Zf

10 *m * R^2 * Zi = 26/9 * m * R^2 * Zf

Zf/Zi = 3.461

the ratio of final to initial angular speed is 3.461

b)

now , for the ratio of initial to final centrifugal force

FCf/FCi = m * Zf^2 * (R/3)/(m * Zi^2 * R)

FCf/FCi = (Zf/Zi)^2/3

FCf/FCi = (45/13)^2/3

FCf/FCi = 3.99

c)

Zi = 1/4 rev/s

Zf/Zi = 3.461

Zf = 3.461/4 rev/s

Zf = 0.865 rev/s

the final angular speed is 0.865 rev/s

d)

for the initial centrifugal force = m * Zi^2 * R

initial centrifugal force = 65 * (0.25 * 2pi)^2 * 2

initial centrifugal force = 320 N

final centrifugal force = 3.99 * 320

final centrifugal force = 1278.5 N

the final centrifugal force is 1278.5 N

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