Question

An electric fan spinning with an angular speed of \(13 \mathrm{rad} / \mathrm{s}\) has a kinetic energy of \(4.6 \mathrm{~J}\). What is the moment of inertia of the fan?

Short answer

The moment of inertia of the fan is approximately 0.0545 kg·m².

Step-by-step solution

Understanding the Relationship

The problem mentions angular speed and kinetic energy. These are related by the rotational kinetic energy formula, given by:\[K = \frac{1}{2} I \omega^2\]where \( K \) is the kinetic energy, \( I \) is the moment of inertia, and \( \omega \) is the angular speed.

Identifying Known Values

From the problem, we know that the kinetic energy \( K = 4.6 \) J and the angular speed \( \omega = 13 \) rad/s. We need to find the moment of inertia \( I \).

Rearranging the Formula

We start with the formula for rotational kinetic energy:\[K = \frac{1}{2} I \omega^2\]To find \( I \), rearrange the formula:\[I = \frac{2K}{\omega^2}\]

Calculating the Moment of Inertia

Substitute the known values into the rearranged formula:\[I = \frac{2 \times 4.6}{13^2}\]Calculate \( 13^2 = 169 \), so:\[I = \frac{9.2}{169} \approx 0.0545 \, \mathrm{kg} \cdot \mathrm{m}^2\]

Key concepts

Angular Speed

Angular speed is a measure of how fast an object rotates or spins around its axis. It tells us how many radians an object covers in a second. Radians are a unit of angle measurement, much like degrees but part of the metric system. This makes them perfect for calculations in physics because they fit seamlessly into the mathematical tools used.

A fan spinning with an angular speed means it rotates at a certain number of radians per second. The faster the fan spins, the higher its angular speed. In our exercise, the fan's angular speed is given as 13 rad/s. Here, "rad" represents radians, and "s" represents seconds. This means every second, the fan sweeps out 13 radians along its circular path.

Understanding angular speed is crucial because it links directly to other rotational properties like rotational kinetic energy. Angular speed is key in determining how energy is distributed in rotating systems, whether we're talking about small table fans or massive space stations. So whenever you hear about something rotating, think about how angular speed plays a part.

Kinetic Energy

Kinetic energy is the energy an object possesses because of its motion. In the realm of physics, it is one of the fundamental types of energy that can be converted into other forms, such as potential energy, and vice versa.

For a moving object not spinning, kinetic energy is given by \[ K = \frac{1}{2} m v^2 \]where \( m \) is mass and \( v \) is velocity. However, when we talk about rotational motion, like that of fans, we use rotational kinetic energy, which depends on angular speed and moment of inertia instead.

• Kinetic energy is directly proportional to the square of speed or velocity. This means if you double the speed, the kinetic energy increases by four times.
• It is always a positive value because it’s an expression of motion, and motion isn’t negative.

In the problem at hand, the fan has a kinetic energy of 4.6 J (joules), a critical component for determining its moment of inertia. Kinetic energy helps us understand how much work is needed to set the object in motion or to keep it moving.

Rotational Kinetic Energy

Rotational kinetic energy is the energy of an object due to its rotational movement. Just like linear kinetic energy relates to linear motion, rotational kinetic energy is concerned with objects that spin or revolve.

The formula for this type of energy is given by \[ K = \frac{1}{2} I \omega^2 \]where:

• \( K \) is the rotational kinetic energy.
• \( I \) is the moment of inertia, which measures how difficult it is to change an object's rotation.
• \( \omega \) is the angular speed.

This formula indicates how energy is distributed in rotating systems based on the system's mass distribution (moment of inertia) and its angular velocity.

For the electric fan in the exercise, the given kinetic energy and angular speed allow us to find the fan's moment of inertia. Rearranging the formula to solve for \( I \), we substitute our known values to reveal essential characteristics of the object's rotational motion. Through this understanding, the abstract concept of energy becomes a tangible number reflecting real-world physics in action.