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Answer :
Final answer:
To solve this, calculate the moment of inertia for the airplane propeller by treating it as a slender rod rotating about its center. Convert the given rotational speed from revolutions per minute to radians per second. Finally, use these values in the kinetic energy formula to obtain the rotational kinetic energy of the airplane propeller as 2.81 × 10⁶ Joules.
Explanation:
The kinetic energy of rotation, represented as K, is given by the formula K = 0.5 * I * ω², where I is the moment of inertia and ω is the angular speed in radians per second. First, we calculate the moment of inertia for the airplane propeller, considering it as a slender rod rotating about its center.
The moment of inertia, I, in this case can be represented by the formula I = (1/12) * m * L², where m is the mass and L is the length of the propeller. Considering the propeller's mass m = 128 kg and its length L = 1.97m, we find I = 65.93 kg-m².
Next, to find the angular speed, ω, we convert the given rotational speed from revolutions per minute to radians per second. As there are 2π radians in one revolution and 60 seconds in a minute, our conversion results in ω = 2800 * (2π/60) rad/s = 292.168 rad/s.
Plugging these values into the kinetic energy formula, we find: K = 0.5 * I * ω² = 0.5 * 65.93 kg-m² * (292.168 rad/s)² = 2.81 × 10⁶ J. Therefore, the rotational kinetic energy of the propeller is 2.81 × 10⁶ Joules.
Learn more about Rotational Kinetic Energy here:
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