Question

The angle turned through by the flywheel of a generator during a time interval \(t\) is given by $$ \phi=a t+b t^{3}-c t^{4}, $$ where \(a, b\), and \(c\) are constants. What is the expression for its (a) angular velocity and ( \(b\) ) angular acceleration?

Short answer

The angular velocity \(w = a + 3bt^{2} - 4ct^{3}\) and the angular acceleration \(\alpha = 6bt - 12ct^{2}\).

Step-by-step solution

Compute the derivative of the angular displacement to find the angular velocity

The angular velocity \(w\) is the derivative of the angular displacement \(\phi\) with respect to time: \(w = \frac{d\phi}{dt}\). Therefore, differentiate the expression for \(\phi\) to obtain: \(w = \frac{d}{dt}(at + bt^{3} - ct^{4}) = a + 3bt^{2} - 4ct^{3}\).

Compute the derivative of the angular velocity to find the angular acceleration

The angular acceleration \(\alpha\) is the derivative of the angular velocity \(w\) with respect to time: \(\alpha = \frac{dw}{dt}\). Therefore, differentiate the expression for \(w\) to obtain: \(\alpha = \frac{d}{dt}(a + 3bt^{2} - 4ct^{3}) = 6bt - 12ct^{2}\).

Key concepts

Angular Velocity

Angular velocity is a measure of how fast an object rotates or spins around a specific axis. Imagine a clock - how fast the hands move around the clock's face is an example of angular velocity. In physics, angular velocity is denoted by \( \omega \), and it is the rate at which the angular position of an object changes with respect to time.

Given an angular displacement \( \phi \) expressed as a function of time \( t \), you can find angular velocity by taking the derivative of \( \phi \) with respect to \( t \). This procedure essentially tells you how quickly the angle is changing at any given moment. For example, as in the exercise, if \( \phi = a t + b t^3 - c t^4 \), then the angular velocity, \( \omega \), is calculated by differentiating this expression:

• \( w = \frac{d}{dt}(at + bt^{3} - ct^{4}) = a + 3bt^{2} - 4ct^{3} \)

Understanding the concept of angular velocity is essential because it provides insight into the rotational motion's dynamic nature, which is crucial in fields ranging from mechanical engineering to physics research.

Angular Acceleration

Angular acceleration describes how quickly the angular velocity of an object changes over time. It is the rotational equivalent of linear acceleration and is denoted by \( \alpha \). You can think of it as the "speeding up" or "slowing down" of rotational motion.

To calculate angular acceleration, you need to differentiate the angular velocity \( \omega \) with respect to time \( t \). This gives you the rate at which the angular velocity itself is changing. In the given problem, we started with the expression for angular velocity \( w = a + 3bt^{2} - 4ct^{3} \) and took its derivative to find:

• \( \alpha = \frac{d}{dt}(a + 3bt^{2} - 4ct^{3}) = 6bt - 12ct^{2} \)

Understanding angular acceleration is key for analyzing systems where objects experience changes in rotational motion, such as wheels speeding up or slowing down, or planets orbiting stars. It helps in predicting the future motion of these systems.

Derivatives in Physics

Derivatives play an essential role in physics, especially in analyzing motion, both linear and angular. They provide a mathematical tool to measure how a particular quantity changes over time. When working with functions that describe physical phenomena, differentiating these functions allows us to understand aspects like velocity, acceleration, and angular motion.

In the context of angular motion:

• Taking the derivative of an angular position function gives the angular velocity.
• Subsequently, differentiating the angular velocity provides the angular acceleration.

By applying the rules of differentiation to physics problems, you gain a deeper understanding of how systems evolve. For instance, the process of deriving the formulas for \( \omega \) and \( \alpha \) demonstrates how derivatives transform complex rotational motion into comprehensible expressions. This approach is a cornerstone of calculus applications in scientific analysis, laying the groundwork for more advanced studies in physics and engineering.