The rigid body shown in the diagram above consists of a vertical support past an

Question

The rigid body shown in the diagram above consists of a vertical support past and two horizontal crossbars with spheres attached. The masses of the spheres and the lengths of the crossbars are indicated in the diagram. The body rotates about a vertical axis along the support post with constant angular speed w. If the masses of the support post and the crossbars are negligible, what is the ratio of the angular momentum Lu/Ld of the two upper spheres to that of the two lower spheres? None of these is correct. Lu/Ld = 50 */147 Lu/Ld = 7 */18 Lu/Ld = 504 */275 Lu/Ld = 2 */1 Lu/Ld = 35 */8 Lu/Ld = 49 */125 Lu/Ld = 16 */13 Lu/Ld = 49 */25 Lu/Ld = 9 */4

Explanation / Answer

Since the radii of the spheres is not given, this means that we treat all of the spheres as points.

The moment of inertia of a point mass is   I = M R2 , where M is the mass of the particle, and R is the distance from the rotating particle to the axis of rotation (ie, the support post).

angular momentum = I by defn

upper angular momentum =   (2) (2 m) ( 7 L)2 , where the factor of 2 is due to the fact that there are two identical masses rotating with angular speed .

down angular momentum = (2) (10m) ( 5 L)2

When taking the ratio of the two above angular momenta, the factors of 2, the distance L, and all cancel out to give:

ratio of   upper/down =   2(49) /[(10)(25)] = 49/125 final answer for the desired ratio.

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