An airplane propeller is 1.1 m in length (from tip to tip) and has a mass of 2.8 kg . When the airplane's engine is first started, it applies a constant torque of 8.1 N⋅m to the propeller, which starts from rest. What is the angular acceleration of the propeller? Treat the propeller as a slender rod. The unit of the angular acceleration is rad/s2
alpha = tau/I
You'll need to look up I for a rod spun around one end.
To find the angular acceleration of the propeller, we can use the following equation:
τ = Iα
where τ is the torque applied to the propeller, I is the moment of inertia, and α is the angular acceleration.
First, let's calculate the moment of inertia of the propeller. Since the propeller is treated as a slender rod, we can use the equation for the moment of inertia of a slender rod rotating about its center:
I = (1/12) * m * L^2
where m is the mass of the propeller and L is the length of the propeller.
Plugging in the values:
m = 2.8 kg
L = 1.1 m
I = (1/12) * 2.8 kg * (1.1 m)^2
Next, we can substitute the values of torque and moment of inertia into the equation to find the angular acceleration:
8.1 N⋅m = I * α
Solving for α:
α = 8.1 N⋅m / I
Now, let's substitute the value of I into the equation and calculate the angular acceleration:
α = 8.1 N⋅m / [(1/12) * 2.8 kg * (1.1 m)^2]
Calculating the final result, we get:
α ≈ 32.65 rad/s²
Therefore, the angular acceleration of the propeller is approximately 32.65 rad/s².