Question

The propeller of a light plane has a length of \(2.092 \mathrm{~m}\) and a mass of \(17.56 \mathrm{~kg}\). The rotational energy of the propeller is \(422.8 \mathrm{~kJ}\). What is the rotational frequency of the propeller (in rpm)? You can treat the propeller as a thin rod rotating about its center.

Short answer

Solution: 1. Calculate the moment of inertia: \(I = \frac{1}{12} \times 17.56 \times (2.092)^2\) 2. Rearrange the formula for rotational energy: \(\omega = \sqrt{\frac{2E}{I}}\) 3. Calculate the angular velocity: \(\omega = \sqrt{\frac{2 \times 422800}{I}}\) 4. Convert the angular velocity to rotational frequency: \(n = \frac{\omega}{2\pi} \times 60\)

Step-by-step solution

Find the moment of inertia of the propeller

To find the moment of inertia of the propeller, we can treat it as a thin rod rotating about its center. The moment of inertia for a thin rod is given by the formula \(I = \frac{1}{12}mL^2\), where m is the mass of the rod and L is its length. Given, mass \(m = 17.56 \mathrm{~kg}\) and length \(L = 2.092 \mathrm{~m}\), we can find the moment of inertia: \(I = \frac{1}{12} \times 17.56 \times (2.092)^2\)

Use the formula for rotational energy

The rotational energy of the propeller is given as \(E = 422.8 \mathrm{~kJ} = 422800 \mathrm{~J}\). The formula for rotational energy is \(E = \frac{1}{2}I\omega^2\), where \(\omega\) is the angular velocity and \(E\) is the energy. We can rearrange this formula to find the angular velocity: \(\omega^2 = \frac{2E}{I} \implies \omega = \sqrt{\frac{2E}{I}}\)

Calculate the angular velocity

Now, we can use the rotational energy and the moment of inertia to calculate the angular velocity: \(\omega = \sqrt{\frac{2 \times 422800}{I}}\) After finding the moment of inertia in step 1, substitute its value in this formula.

Convert the angular velocity to rotational frequency

The angular velocity is given in radians per second, but we need the rotational frequency in revolutions per minute. We can use the following conversion: \(n = \frac{\omega}{2\pi} \times 60\) Where \(n\) is the rotational frequency in revolutions per minute (rpm), \(\omega\) is the angular velocity in radians per second, and the factor \(60\) is used to convert from seconds to minutes. Using the angular velocity calculated in step 3: \(n = \frac{\omega}{2\pi} \times 60\) This will give us the rotational frequency of the propeller in rpm.

Key concepts

Moment of Inertia

The moment of inertia, often represented as 'I', is a measure of an object’s resistance to changes in its rotation rate. It plays a critical role in rotational dynamics, just as mass does in linear motion. For any given object, its moment of inertia depends on the distribution of its mass relative to the axis of rotation.

To find the moment of inertia for different shapes, various formulas are used. For a thin rod rotating about its center, the formula is: \( I = \frac{1}{12}mL^2 \). Here, 'm' signifies the mass of the rod, and 'L' is its length. The moment of inertia is directly proportional to both the mass of the object and the square of its rotational length. This means that the further the mass is distributed from the rotational axis, the greater the moment of inertia, and consequently, the more force it requires to change its rotational speed.

In the case of the propeller exercise, the mass and length of the propeller were given, enabling the calculation of its moment of inertia. This value serves as a foundation to assess the rotational energy and frequency, as it quantifies how much kinetic energy can be stored in the propeller's rotation given its mass distribution.

Rotational Energy

Rotational energy, also known as rotational kinetic energy, is the energy possessed by a rotating object due to its motion. It is analogous to linear kinetic energy but for rotating systems. The formula to calculate rotational energy is \( E = \frac{1}{2}I\omega^2 \), where 'E' represents the rotational energy, 'I' is the moment of inertia, and 'ω' (omega) is the angular velocity.

This formula shows how rotational energy is influenced by both the moment of inertia and the square of the angular velocity. Thus, an object having a high moment of inertia or rotating at a high speed will have substantial rotational energy. For instance, in the exercise with the airplane propeller, knowing the rotational energy and the moment of inertia allows us to solve for the angular velocity. This is vital because it indicates how the propeller's mass distribution and speed contribute to the amount of stored energy. Since the propeller's rotational energy was provided, it could be used to find the angular velocity, crucial for obtaining the rotational frequency.

Angular Velocity

Angular velocity, denoted by 'ω', measures the rate of rotation of an object. It is defined as the angle rotated per unit of time and is typically measured in radians per second (rad/s). In the context of the propeller problem, angular velocity is essential because it determines the speed at which the propeller is rotating.

Angular velocity is related to the kinetic energy of rotation. As the formula \( \omega = \sqrt{\frac{2E}{I}} \) indicates, angular velocity can be calculated if the rotational energy and the moment of inertia are known. Once 'ω' is found, it can be converted into rotational frequency, which is commonly measured in revolutions per minute (rpm). The formula for this conversion is \( n = \frac{\omega}{2\pi} \times 60 \).

The practical application of understanding angular velocity goes beyond exercises. It’s crucial in engineering, where the operational speeds of components are significant for both functionality and safety. For instance, in the propeller exercise, determining the angular velocity would ensure the propeller operates within safe limits while maintaining the effectiveness of the airplane's propulsion system.