Question

A ballet dancer spins about a vertical axis at $24$ r.p.m with outstretched arms folded the M.I. about the same axis changes by $60\%$. The revolution is.
$A)\;60{\text{ }}r.p.m$
$B)15{\text{ }}r.p.m$
$C)40{\text{ }}r.p.m$
$D)17{\text{ }}r.p.m$

Answer 1

We know the relationship between the moment of inertia and angular velocity which helps us to solve this problem.
We can also use the conservation of momentum to solve this problem.
When dancers fold their arms the moment of inertia gets varied. The angular momentum involves the angular velocity and the moment of inertia.

Formula Used:
The angular momentum is given as follows,
$L = I.\omega$
Where,
$I$ is the moment of inertia.
$L$ is the angular momentum
$\omega$is the angular velocity.

Complete step-by-step solution:
Here, initially, the dancer has $24$ rpm, and when arms are stretched the moment of inertia changes to $60\%$ initially, $M.I = I$
$\omega = 24rpm$
The angular momentum is given as follows,
$L = I.\omega$
Where,
$I$ is the moment of inertia.
$L$ is the angular momentum.
$\omega$ is the angular velocity.
After changing it will be, $60\%$
$\dfrac{{60}}{{100}}I{\omega _0} = I\omega$
$\Rightarrow \dfrac{3}{5}{\omega _0} = 24$
$\Rightarrow {\omega _0} = \dfrac{{24 \times 5}}{3}$
After the simplification, ${\omega _0}$is,
${\omega _0} = 40rpm$
angular momentum is said to be momentum or rotational momentum. It is rotational which is equivalent to linear momentum. It is the most important quantity in physics because it gives a quantity the total angular momentum of a closed system remains constant.

Note: Generally, the Angular momentum is an extensive quantity.
The angular momentum of any of the composite systems is computed by the sum of the angular momentum of its constituent parts.
angular momentum per unit volume is zero over the entire body.