Question

An airplane propeller is 2.78 m in length (from tip to tip) and has a mass of 137 kg . When the airplane's engine is first started, it applies a constant torque of 1890N*m to the propeller, which starts from rest.
a)What is the angular acceleration of the propeller? Treat the propeller as a slender rod.
b)What is the propeller's angular speed after making 5.00 rev ?
c)How much work is done by the engine during the first 5.00 rev ?
d)What is the average power output of the engine during the first 5.00 rev ?
e)What is the instantaneous power output of the motor at the instant that the propeller has turned through 5.00 rev ?

Answer 1

Answer:

a) $\alpha=21.42rad/s^2$

b) $\omega f=36.69rad/s$

c) W = 59385.7J

d) P = 34728.5W

e) P = 69344.1W

Explanation:

Our given data are:

τ=1890N.m       M=137kg       L = 2.78m

With these values we can calculate the inertia as:

$I =M/12*L^2=88.23kg.m^2$

We know that the torque is:

$\tau=I*\alpha$    Solving for the acceleration:

$\alpha=\tau/I=21.42rad/s^2$

With kinematics formula we calculate the angular speed:

$\omega f^2=\omega o^2+2*\alpha*\theta$   where ωo=0 and θ=5*2π rad

$\omega f = 36.69rad/s$

The total work done by the engine is equal to the variations on the kinetics energy:

$W = \Delta K = I/2*\omega^2-0=59385.7J$

Agerage power is given by:

$P=W/\Delta t$   where time interval can be found with a kinematic formula as: $t=\omega f/\alpha=1.71s$

So, average power will be:

P = 59385.7 / 1.71 = 34728.5W

Instantaneous power will be:

P = τ*ω = 1890 * 36.69 = 69344.1W

Answer 2

Inertia is the tendency of a body to resist the change in momentum. The angular acceleration of the propeller is $21.42 \rm \ rad/s^2$.  

What is Inertia?

Inertia can be defined as the tendency of objects to resist the change in position or direction of the object.

The inertia can be calculated by the formula,

$I = \dfrac {1}{12}m l^2$

Where,

$m$ - mass = 137 kg

$l$ - length =  2.78m

Put the values,

$I = \dfrac1{12} 137 \times 2.78^2\\\\I = 88.23 \rm / kg m^3$

The angular acceleration can be calculated by torque,

$\tau = Ia\\\\a = \dfrac \tau I$........1

Where,

a - acceleration

$\tau$ - torque = 1890 Nm    

Put the values equation 1,

$a = \dfrac {1890 }{88.23}\\\\a = 21.42 \rm \ rad/s^2$

Therefore, the angular acceleration of the propeller is $21.42 \rm \ rad/s^2$.  

Learn more about angular acceleration:

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