How Does Mass Affect Angular Velocity in Rotational Systems?

[melissa_y]

1) A 5.4m diameter merry-go-round is rotating freely with an angular velocity of 0.730 rad/s. Its total moment of inertia is 1665kg*m2. Four people standing on the ground, each of mass 60.8kg, suddenly step onto the edge of the merry-go-round. What is the angular velocity of the merry-go-round now? Use units of "rad/s".

2)A solid sphere of radius 13.0 cm and mass 12.0 kg starts from rest and rolls without slipping a distance of L = 7.0 m down a house roof that is inclined at 32o. What is the angular speed about its center as it leaves the house roof? Use units of "rad/s".

 

[thepatientmental]

1) When the four people step onto the merry-go-round, the total moment of inertia increases to 1665kg*m2 + 4*(60.8kg)*(5.4m)^2 = 3547.68 kg*m2. By the conservation of angular momentum, the initial and final angular momenta are equal. Therefore, the final angular velocity is given by:
Iω = I'ω'
1665kg*m2 * 0.730 rad/s = 3547.68 kg*m2 * ω'
ω' = (1665kg*m2 * 0.730 rad/s)/3547.68 kg*m2 = 0.343 rad/s

2) The potential energy of the sphere at the top of the roof is converted into rotational kinetic energy as it rolls down the roof, so the final angular speed is given by:
KErot = PE = mgh = (1/2)Iω^2
(1/2)(12.0 kg)(9.8 m/s^2)(7.0 m) = (1/2)(2/5)(12.0 kg)(0.13 m)^2ω^2
ω = (2*9.8*7)/(2/5*0.13^2)^(1/2) = 6.45 rad/s