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 ; disk B has radius ; and disk C has radius 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?
The ratio of the magnitude of the angular momentum of disk to that of disk is
i) The disk has radius
ii) Hub of disk has radius
iii) The disk has radius
iv) The disk has radius
v) Density of disk and diskis same as
Use the expression of angular momentum in terms of rotational inertia and angular velocity. Find the mass of each disk by using the expression of density. All disks are connected by the same belt at the rim as well as its hub; hence their linear velocity will be the same. We use the expression of relation between linear velocity, angular velocity, and their radius.
The linear speed at the rim of disk must equal the linear speed at the rim of disk .
The linear speed at the hub of disk must equal the linear speed at the rim of disk
From equations (i) and (ii) as
The angular momenta depend on their angular velocities as well as their moment of inertia.
The moment of inertia of the disk about an axis and passing through its center is
If is the thickness and is the density of each disk, then
The ratio of angular momenta of disk to the disk is
What happens to the initially stationary yo-yo in Fig. 11-25 if you pull it via its string with (a) force (the line of action passes through the point of contact on the table, as indicated), (b) force (the line of action passes above the point of contact), and (c) force (the line of action passes to the right of the point of contact)?
A Texas cockroach of mass runs counterclockwise around the rim of a lazy Susan (a circular disk mounted on a vertical axle) that has radius 15 cm, rotational inertia , and frictionless bearings. The cockroach’s speed (relative to the ground) is , and the lazy Susan turns clockwise with angular speed The cockroach finds a bread crumb on the rim and, of course, stops.
(a) What is the angular speed of the lazy Susan after the cockroach stops?
(b) Is mechanical energy conserved as it stops?
Two balls are attached to the ends of a thin rod of length role="math" localid="1661007264498" and negligible mass. The rod is free to rotate in a vertical plane without friction about a horizontal axis through its center. With the rod initially horizontal (In Figure), wad of wet putty drops onto one of the balls, hitting it with a speed of and then sticking to it.
(a) What is the angular speed of the system just after the putty wad hits?
(b) What is the ratio of the kinetic energy of the system after the collision to that of the putty wad just before?
(c) Through what angle will the system rotate before it momentarily stops?
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