# Circular Motion

## The force and acceleration vectors point to the center of the circle.

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Spinning on Ice

### Spinning on Ice

Credit: Pietro Zanarini
Source: http://www.flickr.com/photos/7363465@N08/3725912776
License: CC BY-NC 3.0

Performed by every professional figure skater, the objective of the spin is to hold a specific body position while rotating around a single point on the ice.

#### News You Can Use

Credit: deerstop
Source: http://commons.wikimedia.org/wiki/File:Cup_of_Russia_2010_-_Yuzuru_Hanyu_(spin).jpg
License: CC BY-NC 3.0

Yuzuru Hanyu performing a sit spin [Figure2]

• A spin is accomplished by the skater rotating on the part of the blade that is just behind the toe pick. Skaters can perform different types of spins depending on the position of the skater's arms, legs, and body. Spins fall into three different categories: upright spins, sit spins and camel spins. As a skater spins, it can be observed that the rotation speed of the skater generally increases as an outstretched limb is brought closer. This is a result of the conservation of angular momentum.
• Angular momentum is nearly identical to ordinary translational momentum \begin{align*}p\end{align*}, except it deals with velocities that are defined in rotational terms. Angular momentum is defined as:

\begin{align*}\overrightarrow{L} = \overrightarrow{r} \times \overrightarrow{p} = rmv \sin \theta\end{align*}

In the above equation, \begin{align*}\overrightarrow{L}\end{align*} represents the angular momentum vector, \begin{align*}\overrightarrow{r}\end{align*} represents the distance between the object and the point from which it's rotating, and \begin{align*}\overrightarrow{p}\end{align*} represents the translational momentum vector.

• As with translational momentum, angular momentum must be conserved as shown in the following equation.

\begin{align*}\Delta L&= L_{final} - L_{initial}=0 \\ \rightarrow L_{final} &= L_{initial}\end{align*}

So when an ice skater who is spinning with a given angular velocity, \begin{align*}\omega\end{align*}, brings an outstretched limb closer, \begin{align*}r\end{align*} becomes smaller. Since angular momentum must be conserved, the angular velocity \begin{align*}\omega\end{align*} must increase if \begin{align*}r\end{align*} decreases. (Remember that \begin{align*}\omega=v/r\end{align*}.) Therefore, the skater speeds up.

• Learn more about angular momentum below:

#### Show What You Know

Using the information provided above, answer the following questions.

1. When the water bottles were brought closer to Bill's body in the video, why did he speed up?
2. Are the angular velocity and the angular momentum parallel or perpendicular to one another?

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### Image Attributions

1. [1]^ Credit: Pietro Zanarini; Source: http://www.flickr.com/photos/7363465@N08/3725912776; License: CC BY-NC 3.0
2. [2]^ Credit: deerstop; Source: http://commons.wikimedia.org/wiki/File:Cup_of_Russia_2010_-_Yuzuru_Hanyu_(spin).jpg; License: CC BY-NC 3.0

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