Problem
Two lightweight rods 24 cm in length are mounted perpendicular to an axle and at 180° to each other. At the end of each rod is a 480 g mass. The rods are spaced 42 cm apart along the axle. The axle rotates at 4.5 rad/s.
(a) What is the component of the total angular momentum along the axle?
(b) What angle does the vector angular momentum make with the axle?
[Hint: Remember that the vector angular momentum must be calculated about the same point for both masses, which could be the CM.]
Solution
(a)
Assume that the system is rotating counterclockwise when viewed from above along the rod. At the instant shown in the figure, the velocity of mass A is towards the viewer, and the velocity of mass B is away from the viewer. Use a Cartesian coordinate system where the positive x axis to the right, the positive y axis up along the axle, and the positive z axis out of the plane of the diagram towards the viewer. The speed is given by 
(4.5 rad/s)(0.24 m)=1.08 m/s
![{\displaystyle {\vec {L}}={\vec {r}}_{A}\times {\vec {p}}_{A}+{\vec {r}}_{B}\times {\vec {p}}_{B}=m\left[{\vec {r}}_{A}\times {\vec {v}}_{A}+{\vec {r}}_{B}\times {\vec {v}}_{B}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9c9d303777375956afdd9aed6563c0607fe8a905)
![{\displaystyle =m\left[\left|{\begin{array}{ccc}{\hat {i}}&{\hat {j}}&{\hat {k}}\\-0.24{\text{ m}}&0.21{\text{ m}}&0\\0&0&v\end{array}}\right|+\left|{\begin{array}{ccc}{\hat {i}}&{\hat {j}}&{\hat {k}}\\0.24{\text{ m}}&-0.21{\text{ m}}&0\\0&0&-v\end{array}}\right|\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a7f5631ed8790f8141c5a61489bbab286601ed61)
![{\displaystyle =m\left[2{\hat {i}}(0.21({\text{ m}})v+2{\hat {j}}(0.24({\text{ m}})v\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e4c7b7bfa986ff4a15a1d7c71a7a8b7ebf10ad0f)
![{\displaystyle =2mv\left[{\hat {i}}(0.21({\text{ m}})+{\hat {j}}(0.24({\text{ m}})\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5dce9f1d4e96a59aaaa152ebc07ab1c14f886151)
![{\displaystyle =2(0.48({\text{ kg}})(1.08({\text{ m/s}})\left[{\hat {i}}(0.21({\text{ m}})+{\hat {j}}(0.24({\text{ m}})\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5876825b5ec0dc9adf3be64c11173eed5a0ca6db)
![{\displaystyle =\left[{\hat {i}}(0.2177)+{\hat {j}}(0.2488)\right]{\text{ kg.m}}^{2}/{\text{s}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8bdf356c2d9b6563466c6827eba43393807ef3f0)
The component along the axis is the 
component, 0.25 kg.m 
/s.
(b)
The angular momentum vector will precess about the axle. The tip of the angular momentum vector traces out the dashed circle in the diagram.
