Question

A rod rotates in the vertical plane around a horizontal axle. A wheel is free to rotate on the rod, as shown in Figure 11.74. A vertical stripe is painted on the wheel remains vertical. Is the translational angular momentum of the wheel relative to location A zero or non-zero? If non-zero, what is its direction? Is the rotational angular momentum of the wheel zero or non-zero? If non-zero, what is its direction? Consider a similar system, but with the wheel welded to the rod (not free to turn). As the rod rotates clock wise, does the stripe on the wheel remain vertical? Is the translational angular momentum of the wheel relatives to location A zero or non-zero? If non-zero, what is its direction? Is the rotational angular momentum of the wheel zero or non-zero? If non-zero, what is its direction?

Short answer

The direction of the rotational angular momentum and the direction of translational angular momentum is into the page, when the motion of the wheel is in clockwise.

Step-by-step solution

Definition of Angular Momentum.

The rotating inertia of an object or system of objects in motion about an axis that may or may not pass through the object or system is described by angular momentum.

The rotating analogue of linear momentum is angular momentum (also known as moment of momentum or rotational momentum). A closed system's total angular momentum remains constant.

To find the direction of a positive moving charge's magnetic force, point your right thumb in the direction of the velocity (v), your index finger in the direction of the magnetic field (B), and your middle finger in the direction of the resulting magnetic force.

Figure shows a rod rotating in the vertical plane.

Below figures shows a rod rotating in the vertical plane with a wheel attached.

About the concept of term Translational angular momentum.

The term translation angular momentum is associated with the motion of the center of mass of the system. The magnitude and direction of this differ for different choices of the locations. The expression for a translational angular momentum of a wheel with respect to given locationis,

${\overrightarrow{L}}_{trans}={\overrightarrow{r}}_{CM}\times {\overrightarrow{P}}_{tot}$

Here,${\overrightarrow{L}}_{trans,A}$is the translation angular momentum,${\overrightarrow{r}}_{CM}$is the radius vector of the center of mass, and${\overrightarrow{P}}_{tot}$is total translational momentum of the center of mass.

For a given system, the position vectors of center of mass of a wheel${\overrightarrow{r}}_{CM}$with respect to location and total momentum of the wheel${\overrightarrow{P}}_{tot}$both possesses non-zero values. Moreover, the directions of these vectors are perpendicular to each other. Hence, the contribution of cross product of the translational angular momentum expressing yields non-zero value.

The right-hand rule can be used to determine the direction of this translational angular momentum. Curl the figures of your right hand in the direction of the plane's rotational motion, and extend your thumb to indicate the unit vector direction of the translational angular momentum, according to this rule. The unit vector (your right thumb) is pointing into the page if you rotate clockwise.

About the concept of term Rotational angular Momentum.

The term rotational angular momentum is associated with rotation around the center of mass of the system. In general, rotational angular momentum doesn’t need a subscript because it is calculated relative to the center of mass, and this calculation is unaffected by our choice of the point$A$ ,and by motion of the center of mass.

For a given system, the position vectors of center of mass of a wheel,${\overrightarrow{r}}_{CM}$with respect to location center of mass is zero,${\overrightarrow{r}}_A=0$(here reference point itself is the center of mass point, note not a point$A$ ).This yields, zero rotational angular momentum of the wheel. Here, the direction of the rotational angular momentum can’t be assessing. So, it has no direction.

When the wheel welded to the rod, the direction of the translational angular momentum is-

The rod was riveted to the wheel. A vertical stripe runs across the wheel. The wheel's stripe turns clockwise as the rod revolves clockwise. As a result, the wheel's stripe does not remain vertical.

The translational angular momentum of the wheel relative to location $A$is non-zero quantity, because the center of mass of the wheel changes. Corresponding translational angular momentum of the wheel has some value.

Curl the fingers of our right hand in the direction of rotation in the plane, and the direction of translational angular momentum is in the extended thumb, using the right-hand rule. Because the wheel rotates clockwise, the translation angular momentum is directed into the page.

When the wheel welded to the rod, the direction of the rotational angular momentum is-

The wheel's particles move in a circular manner. The rotational angular momentum of the wheel is a non-zero quantity since it is the sum of the angular momentums of the particles.

Curl the fingers of our right hand in the direction of rotational motion in the plane, and the direction of rotational angular momentum is in the extended thumb, using the right hand rule. Because the wheel rotates clockwise, the rotating angular momentum is directed into the page.

When the wheel is turning clockwise, the direction of rotational angular momentum and translational angular momentum is both into the page.