Question

two solid cylindrical disks have equal radii. the first disk is spinning clockwise at 18 rad/s and the second disk is spinning counterclockwise at 4rad/s. the second disk has a mass 1.37 times larger than the first. if both spinning disks are combined to form one disk, they end up rotating at the same angular velocity and same direction. find this angular velocity after combination if clockwise is positive.

Answer 1

Using conservation of angular momentum, the two combined disks end up rotating clockwise with an angular velocity of 5.28 rad/s after combining. Clockwise rotation is considered positive.

Step-by-step explanation:

To solve the mathematical problem completely, we employ the conservation of angular momentum, which states that the total angular momentum before an interaction must equal the total angular momentum after, provided no external torques act on the system.

The situation involves two rotating disks of equal radii where one disk, spinning clockwise at 18 rad/s, slows down when coupled with a second disk spinning counterclockwise at 4 rad/s. To find the angular velocity after combination, we set up the conservation equation:

Linitial = Lfinal,

where L is the angular momentum. Let m be the mass of the first disk, so the mass of the second disk is 1.37m. Using the formula L = Iω, where I is the moment of inertia (I = mr2 for a solid disk) and ω is the angular velocity, we get:

m•r2•(18) + 1.37m•r2•(-4) = (m + 1.37m)•r2•ωfinal.

Cancelling out the common factor r2, simplifying the masses and solving for ωfinal we get:

18 - 1.37•(4) = (1 + 1.37)•ωfinal,

ωfinal = (18 - 5.48) / 2.37,

ωfinal = 12.52 / 2.37,

ωfinal = 5.28 rad/s (clockwise).

The combined disk ends up rotating clockwise with an angular velocity of 5.28 rad/s.