Substrate Record: qid 10094

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qid 10094 · physics

Question: An airplane propeller, of moment of inertia I, is subject to the driving torque N = N_0 (1 + \alphacos\omega_0t), and a frictional torque due to air resistanceN_f= -b\texttheta ֹ . Find its steady state motion.

1. exttheta = exttheta_0 + (b / I)t + [lphaN_0 / (I \omega_0^2 + b \omega_0^3)][I\omega_0sin\omega_0t + b cos \omega_0t]
2. exttheta = exttheta_0 + (N_0 / b)t + [lphaN_0 / (I^2 \omega_0^3 + b^2 \omega_0)][I\omega_0cos\omega_0t - b sin \omega_0t]
3. exttheta = exttheta_0 + (b / N_0)t + [lphaN_0 / (I^2 \omega_0^2 + b \omega_0^3)][b cos \omega_0t - I\omega_0sin\omega_0t]
4. exttheta = exttheta_0 + (b / N_0)t + [lphaN_0 / (I^2 \omega_0^3 + b^2 \omega_0)][b sin \omega_0t - I\omega_0cos\omega_0t]
5. exttheta = exttheta_0 + [lphaN_0 / (I \omega_0 + b^2 \omega_0^2)][b sin \omega_0t - I\omega_0cos\omega_0t]
6. exttheta = exttheta_0 + (N_0 / I)t + [lphaN_0 / (I \omega_0^2 + b^2 \omega_0)][I\omega_0cos\omega_0t + b sin \omega_0t]
7. exttheta = exttheta_0 + (I / N_0)t + [lphaN_0 / (b \omega_0^2 + I \omega_0^3)][I\omega_0sin\omega_0t - b cos \omega_0t]
8. exttheta = exttheta_0 + (N_0^2 / b)t + [lpha / (I \omega_0^3 + b^2 \omega_0^2)][b sin \omega_0t - I\omega_0cos\omega_0t]
9. exttheta = exttheta_0 + (N_0 / b)t + [lphaN_0 / (I^2 \omega_0^3 + b^2 \omega_0)][b sin \omega_0t - I\omega_0cos\omega_0t]
10. exttheta = exttheta_0 + (N_0 / I)t + [lphaN_0 / (b^2 \omega_0^3 + I^2 \omega_0)][b sin \omega_0t - I\omega_0cos\omega_0t]

Our answer: I. exttheta = exttheta_0 + (N_0 / b)t + [lphaN_0 / (I^2 \omega_0^3 + b^2 \omega_0)][b sin \omega_0t - I\omega_0cos\omega_0t] Source quote machine-checked (at mint)

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