A figure skater is spinning at a rate of 1.7 rev/s with her arms outstretched. She then draws her arms in to her chest, reducing her rotational inertia to 62% of its original value. What is her new rate of rotation?
The angular momentum does not change
original rate of rotation times original rotational inertia
=
final rate of rotation times final rotational inertia
To determine the new rate of rotation for the figure skater, we need to understand the concept of conservation of angular momentum. Angular momentum is the product of rotational inertia and angular velocity.
The initial angular momentum of the skater, L₁, is given by L₁ = I₁ * ω₁, where I₁ is the initial rotational inertia and ω₁ is the initial angular velocity.
The final angular momentum of the skater, L₂, is given by L₂ = I₂ * ω₂, where I₂ is the final rotational inertia and ω₂ is the final angular velocity.
According to the law of conservation of angular momentum, the initial and final angular momenta are equal:
L₁ = L₂
So, I₁ * ω₁ = I₂ * ω₂
Since the skater is reducing her rotational inertia to 62% of its original value, we have I₂ = 0.62 * I₁.
Substituting this value into the conservation equation, we get:
I₁ * ω₁ = (0.62 * I₁) * ω₂
Dividing both sides of the equation by I₁ and simplifying, we have:
ω₁ = 0.62 * ω₂
To find the new rate of rotation, we can substitute the given value of the initial angular velocity, ω₁ = 1.7 rev/s, into the equation:
1.7 rev/s = 0.62 * ω₂
Now, we can solve the equation to find ω₂, the new rate of rotation for the skater. Dividing both sides by 0.62:
ω₂ = 1.7 rev/s / 0.62
Calculating the result:
ω₂ ≈ 2.742 rev/s
Therefore, the new rate of rotation for the figure skater, with her arms drawn in to her chest, is approximately 2.742 rev/s.