Question

The propeller of a light plane has a length of \(2.012 \mathrm{~m}\) and a mass of \(17.36 \mathrm{~kg} .\) The propeller is rotating with a frequency of \(3280 . \mathrm{rpm} .\) What is the rotational kinetic energy of the propeller? You can treat the propeller as a thin rod rotating about its center.

Short answer

Question: Calculate the rotational kinetic energy of a propeller with a mass of 17.36 kg and length of 2.012 m, rotating at a frequency of 3280 rpm. Answer: The rotational kinetic energy of the propeller is approximately 97544 Joules.

Step-by-step solution

Calculate the moment of inertia (I)

For a thin rod rotating about its center, the moment of inertia can be calculated using the formula \(I = \frac{1}{12}mL^2\), where \(m\) is the mass and \(L\) is the length of the rod. Given the mass \(m = 17.36\,\text{kg}\) and the length \(L = 2.012\,\text{m}\), we can now calculate the moment of inertia: \(I = \frac{1}{12}(17.36\,\text{kg})(2.012\,\text{m})^2\)

Calculate the angular velocity (ω)

The propeller is rotating at a frequency of 3280 rpm (revolutions per minute). In order to calculate the angular velocity, we need to convert this frequency into radians per second (rad/s). First, we convert this frequency to revolutions per second (Hz) by dividing it by 60. Then, we multiply the result by \(2\pi\) in order to find the angular velocity in rad/s. Frequency in Hz: \(\frac{3280}{60}\,\text{rev/s}\) Angular velocity: \(\omega = 2\pi \times \frac{3280}{60}\,\text{rad/s}\)

Calculate the rotational kinetic energy

Now that we have the moment of inertia and the angular velocity, we can calculate the rotational kinetic energy using the formula \(KE_{rot} = \frac{1}{2}I\omega^2\). Substitute the obtained values into the formula: \(KE_{rot} = \frac{1}{2}\left(\frac{1}{12}(17.36\,\text{kg})(2.012\,\text{m})^2\right)\left(2\pi \times \frac{3280}{60}\,\text{rad/s}\right)^2\) Now, calculate the result to get the rotational kinetic energy: \(KE_{rot} \approx 97544\,\text{J}\) The rotational kinetic energy of the propeller is approximately \(97544\,\text{Joules}\).

Key concepts

Moment of Inertia

The concept of the moment of inertia plays a critical role in understanding how an object's mass is distributed in relation to its axis of rotation. In essence, it's a measure of an object's resistance to changes in its rotational motion. More mass distributed further from the axis means a higher moment of inertia and, thus, more torque is required to change the object's rotational speed.

For common shapes, there are standard formulas to calculate the moment of inertia. For a thin rod rotating about its center, as in the exercise with the plane's propeller, the formula is \(I = \frac{1}{12}mL^2\), where \(m\) represents the mass and \(L\) the length of the rod. This calculation assumes a uniform mass distribution, which simplifies the computation.

Angular Velocity

Angular velocity is a vector quantity that represents the rate of rotation of an object. It's how fast an object rotates or spins around a fixed axis and is usually measured in radians per second (rad/s).

The relation between linear velocity and angular velocity is given by \( v = \omega r\), where \(v\) is the linear velocity, \(\omega\) is the angular velocity, and \(r\) is the radius of the circular path. In the case of the propeller in the exercise, the frequency of rotation was given in revolutions per minute (rpm), a common unit for rotational speed. However, for calculations involving rotational dynamics, we convert it to rad/s by using the formula \(\omega = 2\pi \times \text{Hz}\), where 1 Hz is equivalent to one revolution per second.

Rotational Dynamics

Rotational dynamics is concerned with the forces and torques that cause changes in rotational motion. It's analogous to linear dynamics, where forces cause changes in linear motion. The key equation of rotational dynamics is Newton’s second law for rotation: \(\tau = I\alpha\), where \(\tau\) is torque, \(I\) is the moment of inertia and \(\alpha\) is the angular acceleration.

The rotational kinetic energy is a part of rotational dynamics. It's the kinetic energy due to the rotation of an object and is given by \(KE_{rot} = \frac{1}{2}I\omega^2\). This equation underscores the relationship between an object's inertia, the square of its angular velocity, and its stored energy of motion. In this exercise, by knowing the moment of inertia and the angular velocity of the propeller, we accurately determine the energy in its rotational movement. Precisely calculating this energy is crucial for understanding the propeller's efficiency and the effort required to bring it to a stop.