three rotating, uniform disks that are coupled by belts. One belt runs around the rims of disks A and C. Another belt runs around a central hub on disk A and the rim of disk B. The belts move smoothly without slippage on the rims and hub. Disk A has radius R; its hub has radius 0.4500R; disk B has radius 0.2500R; and disk C has radius 1.750R. Disks B and C have the same density (mass per unit volume) and thickness. What is the ratio of the magnitude of the angular momentum of disk C to that of disk B?

Question

three rotating, uniform disks that are coupled by belts. One belt runs around the rims of disks A and C. Another belt runs around a central hub on disk A and the rim of disk B. The belts move smoothly without slippage on the rims and hub. Disk A has radius R; its hub has radius 0.4500R; disk B has radius 0.2500R; and disk C has radius 1.750R. Disks B and C have the same density (mass per unit volume) and thickness. What is the ratio of the magnitude of the angular momentum of disk C to that of disk B?

I have tried attempting the question several times : end with answers like :
196 and 525.21 ... Please help i am using the formula :
LC/LB =(½)ρπRhRω/(½)ρπRhRω

Answer 1

To find the ratio of the magnitude of the angular momentum of disk C to that of disk B, we need to use the formula for angular momentum and consider the properties of the disks.

The formula for angular momentum is given by: L = Iω, where L is the angular momentum, I is the moment of inertia, and ω is the angular velocity.

Here's how you can approach the problem step by step:

1. Calculate the moment of inertia for each disk. The moment of inertia for a uniform disk is given by I = ½MR^2, where M is the mass and R is the radius.

For Disk A:
IA = ½MARA^2 + ½MAhub(0.4500R)^2

For Disk B:
IB = ½MBRB^2

For Disk C:
IC = ½MCRc^2

2. Determine the relationship between the masses of the disks. Since Disk B and Disk C have the same density and thickness, their masses can be related by the ratio of their radii squared:
(MB/MC) = (RB^2 / RC^2)

3. Express the angular velocity (ω) in terms of the angular velocity of Disk B. Assume they have the same angular velocity.

ωC = ωB

4. Substitute the equations for the moment of inertia and the angular velocity into the formula for angular momentum:

LC / LB = (IC ωC) / (IB ωB)

5. Substitute the expressions for moment of inertia and the relationship between masses into the ratio equation:

LC / LB = [(½MCRc^2) / ½MBRB^2] * (ωC / ωB)

6. Simplify the equation by canceling out common terms:

LC/LB = (MC / MB) * (Rc^2 / RB^2)

7. Finally, substitute the given values into the equation to find the ratio:

LC/LB = (1.750R^2 / 0.2500R^2) * (MCRc^2 / MBRB^2)

Simplify further if needed to obtain the ratio of the magnitude of the angular momentum of disk C to that of disk B.

Note: Make sure you use the correct units for the radius (R) and the subscripts (A, B, C, hub). Double-check your calculations to avoid any errors.