If a trapeze artist rotates twice each second while sailing through the air, and contracts to reduce her rotational inertia to one-third, how many rotations per second will result?

To solve this problem, we need to understand the concept of rotational inertia and its relationship to rotational speed.

Rotational inertia, also known as moment of inertia, is a measure of an object's resistance to changes in its rotational motion. When a body contracts or changes its shape, its rotational inertia can change.

In this case, the trapeze artist contracts to reduce her rotational inertia to one-third of its original value.

Let's denote the original rotational inertia as I₁ and the reduced rotational inertia as I₂.

According to the problem, I₂ = (1/3) * I₁.

Now, let's assume that the original rotational speed of the trapeze artist is n₁ rotations per second. We want to find the new rotational speed, denoted as n₂.

The conservation of angular momentum states that the product of rotational inertia and rotational speed is constant. In other words, I₁ * n₁ = I₂ * n₂.

Plugging in the values, we have I₁ * n₁ = (1/3) * I₁ * n₂.

Simplifying the equation, we get n₂ = n₁ * (3/1).

Therefore, the trapeze artist will rotate three times per second after contracting.

To summarize:
- Original rotational speed (n₁) = 2 rotations per second
- Reduced rotational speed (n₂) = n₁ * (3/1) = 2 * (3/1) = 6 rotations per second

According to the Law of conservation of angular momentum

L1= L2,
I1•ω1=I2•ω2.
ω1=2πn1
ω2=2πn2
I2=I1/3.
Then n2=(I1/I2)n1=3•2=6 s-1