A ballet student with her arms and a leg extended spins with an initial rotational speed of 0.90rev/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 4.1rev/s .
Determine her initial rotational inertia.
You have to convert rev to rad
To determine the initial rotational inertia of the ballet student, we need to use the principle of conservation of angular momentum. The angular momentum is conserved when no external torques act on the system. The formula for angular momentum is:
L = I ω
Where L is the angular momentum, I is the rotational inertia, and ω is the rotational velocity.
In the initial state, the student has an initial rotational speed of 0.90 rev/s. We can convert this to radians per second by multiplying it by 2π (since 1 revolution is equal to 2π radians).
Initial rotational velocity (ω) = 0.90 rev/s * 2π rad/rev = 5.67 rad/s
The final rotational velocity is given as 4.1 rev/s, which can be converted to radians per second using the same method:
Final rotational velocity (ω) = 4.1 rev/s * 2π rad/rev = 25.76 rad/s
Using the principle of conservation of angular momentum, we can set the initial angular momentum equal to the final angular momentum:
Initial angular momentum (L) = Final angular momentum (L)
I_initial * ω_initial = I_final * ω_final
Since we are trying to find the initial rotational inertia (I_initial), we can rearrange the formula:
I_initial = (I_final * ω_final) / ω_initial
Plugging in the given values:
I_initial = (0.80 kg⋅m² * 25.76 rad/s) / 5.67 rad/s
Calculating:
I_initial = 3.63 kg⋅m²
Therefore, the student's initial rotational inertia is 3.63 kg⋅m².
Io= 3.2 kgm2
When setting Lo=Lf
IoWo=IfWf
Because momentum inertia is conserved in a system
I (final) = I(initial) = 0.8kg * m^2