An airplane propeller is 2.08 m in length (from tip to tip) and has a mass of 117 kg. When the airpline's engine is first started, it applies a constant torque of 1950\ N\cdot m to the propeller, which starts from rest. a) What is the angular acceleration of the propeller? Model the propeller as a slender rod. b) What is the propeller's angular speed after making 5.00 revolutions? c) How much work is done by the engine during the first 5.00 revolutions? e) What is the instantaneous power output of the motor at the instant that the propeller has turne through 5.00 revolutions?

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An airplane propeller is 2.08 m in length (from tip to tip) and has a mass of 117 kg. When the airpline's engine is first started, it applies a constant torque of $$\displaystyle{1950}\ {N}\cdot{m}$$ to the propeller, which starts from rest.
a) What is the angular acceleration of the propeller? Model the propeller as a slender rod.
b) What is the propeller's angular speed after making 5.00 revolutions?
c) How much work is done by the engine during the first 5.00 revolutions?
e) What is the instantaneous power output of the motor at the instant that the propeller has turne through 5.00 revolutions?

2021-02-21
a.
apply the formula for Torque $$\displaystyle{T}={I}\alpha$$
where alpha is Angular accleration
I is moment of inertia $$\displaystyle{\frac{{={M}{L}^{{2}}}}{{{12}}}}$$
$$\displaystyle\alpha={\frac{{{T}}}{{{I}}}}$$
$$\displaystyle\alpha={\frac{{{1950}}}{{{117}\cdot{2.08}\cdot{\frac{{{2.08}}}{{{12}}}}}}}$$
$$\displaystyle\alpha={46.22}\ {r}{a}\frac{{d}}{{s}^{{2}}}$$
b. apply the relation between angular speed and ang accleration
As $$\displaystyle{W}^{{2}}={2}\alpha\theta$$
$$\displaystyle{W}^{{2}}={2}\cdot{46.22}\cdot{5}\cdot{2}\cdot{3.14}$$
$$\displaystyle{W}^{{2}}={2902.616}$$
$$\displaystyle{W}={53.87}\ {r}{a}\frac{{d}}{{\sec{}}}$$
c.apply the formula for Work Done W = 0.5 I $$\displaystyle{W}^{{2}}$$
$$\displaystyle{W}={0.5}\cdot{117}\cdot{2.08}\cdot{2.08}\cdot{53.87}\cdot{\frac{{{53.87}}}{{{12}}}}$$
$$\displaystyle{W}={6.12}\cdot{10}^{{4}}$$ Joules
Power $$\displaystyle{P}={T}{W}$$
so
$$\displaystyle{P}={1959}\cdot{53.87}$$
$$\displaystyle{P}={1.05}\cdot{10}^{{5}}$$ watts
Pin=W/t=1.05 e 5 Watts

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