Use conservation of angular momentum. Assume the outstretched arms have a moment of inertia contribution of
Iarms2 = (1/3) M(arms) L^2, where L = 1.80m m.
[(Ibody + Iarms1)w1] = [(Ibody + Iarms2)w2]
Solve for w2
The moment of inertia about the axis of rotation of the remainder of his body is constant and equal to 0.350 kg*m^2 If the skater's original angular speed is 0.450 rev/s, what is his final angular speed?
Iarms2 = (1/3) M(arms) L^2, where L = 1.80m m.
[(Ibody + Iarms1)w1] = [(Ibody + Iarms2)w2]
Solve for w2
The initial angular momentum (L_initial) of the skater can be calculated using the formula:
L_initial = I_initial * ω_initial
Where:
L_initial is the initial angular momentum
I_initial is the initial moment of inertia of the system
ω_initial is the initial angular speed
The final moment of inertia (I_final) can be obtained by considering the moment of inertia of the skater's arms and the constant moment of inertia of the rest of his body:
I_final = I_arms + I_body
The final angular momentum (L_final) is then given by:
L_final = I_final * ω_final
According to the conservation of angular momentum, L_initial = L_final. So we can equate the two equations and solve for the final angular speed (ω_final).
Let's substitute the given values into the equations:
I_initial = 0.350 kg*m^2 (given)
ω_initial = 0.450 rev/s (given)
I_arms = ½ * m_arms * r^2 = ½ * m_arms * (0.25 m)^2 (moment of inertia of a thin-walled hollow cylinder)
I_body = constant moment of inertia, given as 0.350 kg*m^2
m_arms = 8.50 kg (given)
Substituting these values into the equations, we have:
L_initial = I_initial * ω_initial
L_initial = (0.350 kg*m^2) * (0.450 rev/s)
L_final = I_arms + I_body
L_final = (½ * m_arms * (0.25 m)^2) + 0.350 kg*m^2
Since L_initial = L_final, we can equate the two equations:
(0.350 kg*m^2) * (0.450 rev/s) = (½ * m_arms * (0.25 m)^2) + 0.350 kg*m^2 * ω_final
Now we can solve for ω_final:
ω_final = [(0.350 kg*m^2) * (0.450 rev/s) - (½ * m_arms * (0.25 m)^2)] / (0.350 kg*m^2)
After substituting the given values and calculating this expression, we can find the final angular speed (ω_final).