A figure skater is spinning at a rate of 1.1 rev/s with her arms outstretched. She then draws her arms in to her chest, reducing her rotational inertia to 66% of its original value. What is her new rate of rotation?
(1.1/(.66*1.1))*1.1=1.67
To find the new rate of rotation, we need to apply the conservation of angular momentum principle, which states that the initial angular momentum of a system will remain constant if no external torque is applied.
In this scenario, when the skater draws her arms in, the rotational inertia decreases, but no external torque is involved. Therefore, the initial angular momentum should be equal to the final angular momentum.
The angular momentum of a rotating body can be calculated by multiplying its rotational inertia (I) by its angular velocity (ω). Mathematically, it can be written as:
L = I * ω
Let's denote the initial angular velocity as ω_i and the final angular velocity as ω_f. And let's denote the initial rotational inertia as I_i and the final rotational inertia as I_f.
Initially, the skater spins at a rate of 1.1 revolutions per second (rev/s), which can be converted to angular velocity using the conversion factor 2π radians = 1 revolution. Therefore:
ω_i = 1.1 rev/s * 2π radians/rev = 2.2π radians/s
We also know that the final rotational inertia (I_f) is 66% of the initial rotational inertia (I_i):
I_f = 0.66 * I_i
Since the rotational inertia depends on the mass distribution of the skater's body, which changes when she draws her arms in, we do not have specific values for I_i or I_f. However, we can continue solving the problem by using the given information.
Since angular momentum is conserved, we can write:
I_i * ω_i = I_f * ω_f
Substituting the expressions we found earlier:
(0.66 * I_i) * (2.2π radians/s) = I_f * ω_f
Simplifying the equation:
0.66 * 2.2π * I_i = I_f * ω_f
Dividing both sides by I_f:
0.66 * 2.2π * I_i / I_f = ω_f
Finally, we can plug in the values we have to find the new rate of rotation. Keep in mind that the final angular velocity will depend on the proportions of the skater's arms and body mass distribution, which we don't have specific information about.
By substituting the given values into the equation, you can calculate the new rate of rotation.