A propeller is modeled as five identical uniform rods extending radially from its axis. The length and mass of each rod are 0.721 m and 2.55 kg, respectively. When the propellor rotates at 561 rpm (revolutions per minute), what is its rotational kinetic energy?
Why did the scarecrow become a propeller expert? Because he was outstanding in his field! Now, let's calculate the rotational kinetic energy of the propeller.
First, let's find the total mass of the propeller. Since each rod has a mass of 2.55 kg, and there are five rods, the total mass of the propeller is 5 * 2.55 kg = 12.75 kg.
Next, let's convert the RPM to radians per second. One revolution is equal to 2π radians, so 561 RPM is equal to 561 * 2π / 60 = 586.98 radians per second.
The moment of inertia of the propeller can be calculated using the parallel axis theorem, assuming the rods are thin and rotating perpendicular to their length. The moment of inertia of a uniform rod rotating about one end is given by (1/3) * mass * length^2. Since there are five rods, the moment of inertia for the entire propeller is (1/3) * 12.75 kg * (0.721 m)^2 * 5 = 6.864 kg * m^2.
Finally, we can calculate the rotational kinetic energy using the formula: rotational kinetic energy = (1/2) * moment of inertia * angular velocity^2. Substituting the values, we get rotational kinetic energy = (1/2) * 6.864 kg * m^2 * (586.98 rad/s)^2.
Now, that's a lot of numbers! Let me calculate that for you... *beep boop*
The rotational kinetic energy of the propeller is approximately 12,383.38 Joules. Keep in mind, though, this answer assumes each rod is identical and symmetrically distributed. It's all fun and games until you lose a rod!
To calculate the rotational kinetic energy of the propeller, we need to know its moment of inertia.
The moment of inertia of an object rotating about an axis is given by the equation:
I = Σmiri²
where I is the moment of inertia, m is the mass of each rod, and r is the distance from the axis of rotation.
In this case, we have five identical rods, so we can calculate the total moment of inertia by multiplying the moment of inertia of a single rod by 5:
I_total = I_single * 5
The moment of inertia of a single rod can be calculated as follows:
I_single = (1/3) * m * r²
where r is the length of the rod divided by 2, since it extends radially from the axis.
Let's now substitute the given values into the equations:
r = 0.721 m / 2 = 0.3605 m
m = 2.55 kg
I_single = (1/3) * 2.55 kg * (0.3605 m)²
≈ 0.214 kg·m²
I_total = 0.214 kg·m² * 5
≈ 1.07 kg·m²
Now, let's calculate the rotational kinetic energy using the formula:
K_rotational = (1/2) * I * ω²
where ω is the angular velocity in rad/s. To convert the given rotational speed in rpm to rad/s, we need to multiply by 2π/60:
ω = 561 rpm * (2π rad/rev) / (60 s/min)
= 58.78 rad/s
K_rotational = (1/2) * 1.07 kg·m² * (58.78 rad/s)²
≈ 1840.7 J
Therefore, the rotational kinetic energy of the propeller is approximately 1840.7 Joules.
To find the rotational kinetic energy of the propeller, we need to know its moment of inertia and its angular velocity.
The moment of inertia of an object depends on its shape and mass distribution. In this case, the propeller is modeled as five identical uniform rods extending radially from its axis. We can calculate the moment of inertia of a single rod about its center using the formula for the moment of inertia for a rod rotating about its center:
I = (1/12) * m * l^2
Where:
I = moment of inertia
m = mass of the rod
l = length of the rod
Since the propeller has five identical rods, we can calculate the total moment of inertia of the propeller using the parallel axis theorem, which states that the moment of inertia of an object rotating about an axis parallel to and offset from the axis through its center of mass is equal to the sum of the moment of inertia about its center of mass and the product of its mass and the square of the distance between the two axes:
I_total = I_rod + m * d^2
Where:
I_total = total moment of inertia of the propeller
I_rod = moment of inertia of a single rod about its center
m = mass of the propeller (sum of the masses of the rods)
d = distance between the axis of rotation and the axis through the center of mass of the propeller
Since the propeller is symmetric and rotating about its axis, the axis of rotation coincides with the axis through its center of mass. Therefore, d is zero and the total moment of inertia is equal to the sum of the individual moments of inertia:
I_total = 5 * I_rod
To find the rotational kinetic energy, we can use the formula:
KE_rotational = (1/2) * I_total * ω^2
Where:
KE_rotational = rotational kinetic energy
I_total = total moment of inertia of the propeller
ω = angular velocity (in radians per second)
First, let's calculate the moment of inertia of a single rod:
m = 2.55 kg
l = 0.721 m
I_rod = (1/12) * m * l^2
Next, we calculate the total moment of inertia of the propeller:
I_total = 5 * I_rod
To convert the angular velocity from revolutions per minute (rpm) to radians per second, we use the conversion factor:
1 rpm = (2π/60) rad/s
ω = 561 rpm * (2π/60) rad/s
Now, we can substitute the values into the rotational kinetic energy formula to find the answer.