Question

Two disks are mounted on frictionless bearings on the same axle and can be brought together so that they couple and rotate as one unit. The first disk, which has a mass of 1 kg and radius of 2 m, is set spinning at 45 rad/s. The second disk, which has a mass of 2 kg and a radius of 3 m, is set spinning at 25 rad/s in the opposite direction. They then couple together. What is their angular speed after coupling

Answer 1

12.27 rad/s

Step-by-step explanation:

Moment of inertia = mass x radius^2

For disk 1 = 1 x 2^2 = 4 kg-m^2

For disk 2 = 2 x 3^2 = 18 kg-m^2

Rotational momentum = moment of inertia x angular speed

For disk 1 = 4 x 45 = 180 rad-kg-m/s

For disk 2 = 18 x (-25) = -450 rad-kg-m/s.

Total rational momentum of the system = 180 - 450 = -270 rad-kg-m/s.

The minus means the total rotational momentum is 270 rad-kg-m/s in the direction of disk 2.

According to conservation of angular momentum, initial momentum of system must equal the final momentum of system.

Final momentum of system = total moment of inertia of the system times the new angular velocity of system.

= (4 + 18) x Wf = 22Wf

Equating both moment we have,

22Wf = 270

Wf = 270/22 = 12.27 rad/s

Answer 2

ω3 = 31.15 rad/s

Step-by-step explanation:

given data

mass = 1 kg

radius = 2 m

spinning = 45 rad/s

mass = 2 kg

radius = 3 m

spinning = 25 rad/s

solution

we get here first moment of inertia that is express as

I(1) = 0.5 × M1 × r1² .............1

put here value that is

I(1) = 0.5 × 2 × 2² = 4

and

moment of inertia of disk 2nd is

I(2) = 0.5 × M2 × r2² .............2

I(2) = 0.5 × 2 × 3² = 9

so we get here angular momentum that is express as

I(1) ω1 + I(1) ω2 = ( I(1) + I(2) ) ω3

put here value and we get

4 × 45 + 9 × 25 = ( 4 +9 ) ω3

ω3 = 31.15