Question

A dancer on ice spins faster, when she folds her arms. This is due to

• A. decrease in energy and increase in angular momentum
• B. increase in energy and increase in angular momentum
• C. increase in energy and decrease in angular momentum
• D. constant angular momentum and increase in KE.

Answer 1

The correct option is A decrease in energy and increase in angular momentum

Concept 1

Inertia $I$ of a discrete the mass $m$ at distance $r$ from an axis is given by (about the axis)

$I=m{r}^2$

$\Rightarrow$ $I\propto r^2$

So, when $r$ decreases inertia $I$ decreases

Concept 2

Angular momentum ($L$) is a conserved quantity and remains constant when external. Torque on a body is zero

$\frac{d\overrightarrow{L}}{dt}=\overrightarrow{\tau }$

if $\tau =0$

$\Rightarrow \overrightarrow{L}=$ constant

we know that

$\left|\overrightarrow{L}\right|=I.\omega \Rightarrow \omega =\frac{\left|\overrightarrow{L}\right|}{I}\Rightarrow \omega \propto \frac{1}{I}$

Now

when the dancer folds her arms basically whe is decreasing her inertia

And,

Since $\omega \propto \frac{1}{I}$

So, decreasing of inertia (I), results, in inrease of angular velocity hence

The faster spin faster due to constant momentum and increase in angular velocity