Chapter 12, Problem 11A

The text says that angular momentum is 10, then says it is mvr. Which is it?

To determine

To Explain:The different forms of the angular momentum mathematically.

Explanation of Solution

Introduction:

Spinning of top is the best example of rotational dynamics. In rotational dynamic motion of the body represented by the angular momentum. Angular momentum is the vector quantity and denoted by symbol L . Angular momentum of the body about the center of rotation is,

  $\overrightarrow{L}=\overrightarrow{r}\times \overrightarrow{p}$

Where,

  $\overrightarrow{r}$ is the position vector of the body from the axis of rotation.

  $\overrightarrow{p}$ is the linear momentum of the body.

Suppose body is moving along the circular path having velocity $\overrightarrow{v}$ . $\overrightarrow{r}$ is the radius vector. The direction of the velocity $\overrightarrow{v}$ is along the tangent to the circular trajectory.

  $\begin{array}{l}\overrightarrow{L}=\overrightarrow{r}\times \overrightarrow{p}\\ L=\left|\overrightarrow{r}\right|\cdot \left|\overrightarrow{p}\right|\sin \left(\theta \right)\\ L=\left|\overrightarrow{r}\right|\cdot \left|m\overrightarrow{v}\right|\sin \left(\theta \right)\\ L=mvr\mathrm{....}\left(1\right)\end{array}$

Similarly, for the rigid body rotating about an axis, particle of the rigid body moving along the circle of the radius r with angular velocity $\omega$ .

Linear velocity due to the motion of the particle is,

  $v=r\omega$

From equation (1), Angular momentum can be written as,

  $L=m{r}^2\omega =I\omega \mathrm{...}\left(2\right)$

Where,

  $I=m{r}^2$ is the moment of inertia of body about the axis of rotation.

Conclusion: Equation (1) and equation (2) are the same but represents differently. Depending on the situation of the problem equation (1) or (2) can be used.