The propeller of a light plane has a length of 2.012 m and a mass of 17.36 kg. The propeller is rotating with a frequency of 3280. rpm. What is the rotational kinetic energy of the propeller? You can treat the propeller as a thin rod rotating about its center.
First, we need to convert the frequency into angular velocity (ω) in radians per second.
ω = (3280 rpm) * 2π/60 = 344.44 rad/s
Next, we can use the formula for rotational kinetic energy:
KE = ½ Iω²
Where I is the moment of inertia. For a thin rod rotating about its center, the moment of inertia is:
I = (1/12) M L²
Where M is the mass of the rod, and L is the length.
Plugging in the values, we get:
I = (1/12) * 17.36 kg * (2.012 m)² = 0.906 kg m²
Now we can calculate the kinetic energy:
KE = ½ * 0.906 kg m² * (344.44 rad/s)² = 52,565 J
Therefore, the rotational kinetic energy of the propeller is approximately 52,565 J.
To find the rotational kinetic energy of the propeller, we can use the formula:
Rotational kinetic energy (KE) = (1/2) * moment of inertia * angular velocity^2
First, let's calculate the moment of inertia of the propeller. Since it is treated as a thin rod rotating about its center, the moment of inertia can be calculated using the formula:
moment of inertia (I) = (1/12) * mass * length^2
Given that the length of the propeller is 2.012 m and the mass is 17.36 kg, we can substitute these values into the formula:
I = (1/12) * 17.36 kg * (2.012 m)^2
Calculating this, we get:
I ≈ 0.299 kg * m^2
Next, we need to convert the frequency of rotation from rpm (revolutions per minute) to rad/s (radians per second). Since 1 revolution is equal to 2π radians, we can use the conversion:
angular velocity (ω) = 2π * frequency / 60
Given that the frequency is 3280 rpm, we can substitute this value into the formula:
ω = 2π * 3280 rpm / 60
Calculating this, we get:
ω ≈ 344.44 rad/s
Now, we can substitute the values of the moment of inertia and angular velocity into the formula for rotational kinetic energy:
KE = (1/2) * 0.299 kg * m^2 * (344.44 rad/s)^2
Calculating this, we get:
KE ≈ 8,134.52 Joules
Therefore, the rotational kinetic energy of the propeller is approximately 8,134.52 Joules.