The figure shows 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 0.494R; disk B has radius 0.216R; and disk C has radius 1.64R. 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?
To find the ratio of the magnitude of the angular momentum of disk C to that of disk B, we need to first understand the concept of angular momentum and how it is calculated.
Angular momentum is a property of rotating objects and is defined as the product of the moment of inertia and the angular velocity. The moment of inertia depends on the mass distribution of the object and the axis of rotation, while the angular velocity is the rate at which the object rotates.
In this case, we have three rotating disks coupled by belts. Let's denote the angular momentum of disk C as Lc and the angular momentum of disk B as Lb.
The moment of inertia of a uniform disk rotating about its axis is given by 1/2 * M * R^2, where M is the mass of the disk and R is its radius.
Since disks B and C have the same density and thickness, we can assume they have the same mass per unit area. Therefore, the mass of disk B, Mb, is proportional to its radius (0.216R) squared, and the mass of disk C, Mc, is proportional to its radius (1.64R) squared.
Now, let's consider the angular velocities of the disks. The angular velocity of disk C will be the same as disk B since they are coupled by a belt. So, we can denote the angular velocity of both disks as ω.
Now, we can calculate the angular momenta of disks C and B:
Lc = Ic * ω
Lb = Ib * ω
Substituting the formulas for moment of inertia mentioned earlier:
Lc = (1/2) * Mc * Rc^2 * ω
Lb = (1/2) * Mb * Rb^2 * ω
Since Mc is proportional to (1.64R)^2, and Mb is proportional to (0.216R)^2, we can rewrite the formulas as:
Lc = (1/2) * kc * R^2 * ω
Lb = (1/2) * kb * R^2 * ω
Where kc and kb are constants.
Now, to find the ratio of Lc to Lb, we can divide Lc by Lb:
Lc/Lb = [(1/2) * kc * R^2 * ω] / [(1/2) * kb * R^2 * ω]
The R^2 and ω cancel out, simplifying the expression:
Lc/Lb = kc / kb
Therefore, the ratio of the magnitude of the angular momentum of disk C to that of disk B is simply the ratio of kc to kb. Unfortunately, without specific information about kc and kb, we cannot determine the exact numerical value of this ratio.