Problem 35 - Applied Physics

Question

Find the angular acceleration of a radiator fan blade as its angular speed increases from \(8.50 \mathrm{rad} / \mathrm{s}\) to \(15.4 \mathrm{rad} / \mathrm{s}\) in \(5.20 \mathrm{~s}\).

Step-by-Step Solution

Answer

The angular acceleration is 1.33 rad/s².

Step 1: Understand the Given Data

We are given the initial angular speed \( \omega_i = 8.50 \, \text{rad/s} \), the final angular speed \( \omega_f = 15.4 \, \text{rad/s} \), and the time over which the change occurs \( \Delta t = 5.20 \, \text{s} \).

Step 2: Recall the Formula for Angular Acceleration

Angular acceleration (\( \alpha \)) is the rate of change of angular velocity over time. It can be calculated using the formula \(\alpha = \frac{\omega_f - \omega_i}{\Delta t}\).

Step 3: Substitute the Given Values into the Formula

Plug the given values into the formula for angular acceleration:\[\alpha = \frac{15.4 \, \text{rad/s} - 8.50 \, \text{rad/s}}{5.20 \, \text{s}}\].

Step 4: Calculate the Angular Acceleration

Perform the arithmetic to find angular acceleration:\[\alpha = \frac{6.9 \, \text{rad/s}}{5.20 \, \text{s}} = 1.33 \, \text{rad/s}^2\].

Key Concepts

Angular Velocity

Angular velocity is a measure of how fast something is rotating. It is expressed in radians per second, symbolized as \( \omega \). Imagine a point on the tip of a spinning fan blade. The angular velocity is how fast this point moves around the center of rotation. The faster it goes, the higher the angular velocity.
For example, if a radiator fan blade changes its speed from \(8.50 \text{ rad/s}\) to \(15.4 \text{ rad/s}\) over a few seconds, it shows a change in angular velocity. This tells us how quickly the fan blade is spinning. Here’s what to keep in mind about angular velocity:

• It is a vector quantity, meaning it has both magnitude and direction. The direction is often given by the right-hand rule, using the rotational axis.
• It differs from linear velocity, which measures movement along a straight line, while angular velocity measures rotation.

Understanding angular velocity helps in analyzing any situation involving rotational motion, from simple engines to elaborate satellite systems.

Angular Kinematics

Angular kinematics is the study of rotating objects, describing their motion without considering the forces that cause the motion. It's analogous to linear kinematics but deals with angles and rotations. In the example with the radiator fan, we used angular kinematics to solve for angular acceleration, which shows how the rotational speed changes over time.
Key elements include:

• Initial angular velocity \( \omega_i \)
• Final angular velocity \( \omega_f \)
• Time \( \Delta t \)
• Angular acceleration \( \alpha \) is calculated by \( \alpha = \frac{\omega_f - \omega_i}{\Delta t} \).

Angular kinematics is essential for designing anything that rotates or spins. For instance, engineers need to know these to ensure the functionality and safety of machinery and devices like fans, turbines, and wheels.

Rotational Motion

Rotational motion refers to the motion of an object around a central point or axis. Much like the planets rotate around the Sun, objects in everyday life exhibit rotational motion when they spin. This behavior is governed by several physical quantities, with angular velocity and angular acceleration being two crucial parts.
Understanding rotational motion involves several other concepts as well:

• Torque, which is the force that causes an object to rotate.
• Moment of inertia, which is the rotational mass and depends on the object's shape and axis of rotation.
• Conservation of angular momentum, indicating that in the absence of external torque, an object's angular momentum remains constant.

Studying rotational motion helps us understand and predict the behavior of rotating systems. It's one reason why breaks in a car are so effective, transferring torque to slow the rotation. Mastery of these concepts allows for breakthroughs in areas like mechanical engineering and physics. Great examples can be seen in roller coasters or space stations, where rotational dynamics are crucial for operation and safety.