A ballet student with her arms and a leg extended spins with an initial rotational speed of 1.2 rev/s . As she draws her arms and leg in toward her body, her rotational inertia becomes 0.80 kg⋅m2 and her rotational velocity is 3.9 rev/s .Determine her initial rotational inertia.
To determine the initial rotational inertia, we can use the principle of conservation of angular momentum. According to this principle, the angular momentum of a system remains constant if no external torques are acting on it.
The equation for angular momentum is L = Iω, where L is the angular momentum, I is the rotational inertia, and ω is the angular velocity.
We can write the angular momentum equation for the initial state of the ballet student as:
L_initial = I_initial * ω_initial
Let's solve for I_initial.
Given:
ω_initial = 1.2 rev/s
ω_final = 3.9 rev/s
I_final = 0.80 kg⋅m^2
We need to convert the rotational speeds from revolutions per second to radians per second, as the standard unit for angular velocity is radian per second. We know that 1 revolution is equal to 2π radians.
So, we have:
ω_initial = 1.2 rev/s * 2π rad/rev = 7.54 rad/s
ω_final = 3.9 rev/s * 2π rad/rev = 24.46 rad/s
Now we can substitute the values into the angular momentum equation:
L_initial = I_initial * ω_initial
L_initial = I_final * ω_final (since angular momentum is conserved)
Now we can solve for I_initial:
I_initial = L_initial / ω_initial
I_initial = (I_final * ω_final) / ω_initial
Substituting the given values:
I_initial = (0.80 kg⋅m^2 * 24.46 rad/s) / 7.54 rad/s
Calculating:
I_initial ≈ 2.61 kg⋅m^2
Therefore, the initial rotational inertia of the ballet student is approximately 2.61 kg⋅m^2.