Problem 13 - University Physics with Modern Physics

Question

A turntable rotates with a constant 2.25 \(\mathrm{rad} / \mathrm{s}^{2}\) angular acceleration. After 4.00 \(\mathrm{s}\) it has rotated through an angle of 60.0 \(\mathrm{rad}\) . What was the angular velocity of the wheel at the beginning of the \(4.00-\mathrm{s}\) interval?

Step-by-Step Solution

Answer

The initial angular velocity was 10.5 rad/s.

Step 1: Understand the Given Data

We are given the following information: the angular acceleration is \( \alpha = 2.25 \, \mathrm{rad/s^2} \); the time period is \( t = 4.00 \, \mathrm{s} \); and the angular displacement over the time is \( \theta = 60.0 \, \mathrm{rad} \). We need to find the initial angular velocity \( \omega_0 \).

Step 2: Apply the Angular Motion Equation

Use the angular motion equation: \( \theta = \omega_0 t + \frac{1}{2} \alpha t^2 \). This formula helps us relate angle, initial angular velocity, angular acceleration, and time.

Step 3: Plug in the Known Values

Substitute the known values into the equation: \( 60.0 = \omega_0 (4.00) + \frac{1}{2}(2.25)(4.00)^2 \).

Step 4: Simplify the Equation

Calculate the square of the time and the multiplication: \( 4.00^2 = 16.00 \) and \( \frac{1}{2}(2.25)(16.00) = 18.0 \). This gives us \( 60.0 = 4.00\omega_0 + 18.0 \).

Step 5: Solve for Initial Angular Velocity

Rearrange the equation for \( \omega_0 \): subtract 18.0 from 60.0 to get \( 42.0 = 4.00 \omega_0 \). Divide by 4.00 to solve for \( \omega_0 \): \( \omega_0 = 10.5 \, \mathrm{rad/s} \).

Key Concepts

Angular Acceleration

Angular acceleration refers to how quickly the angular velocity of an object changes over time. In simpler terms, it's the rate at which something speeds up or slows down as it spins. The unit of measurement for angular acceleration is radians per second squared \( \mathrm{rad/s^2} \), which is a standard unit in rotational motion.

In our original exercise, the turntable is experiencing a constant angular acceleration of \(2.25 \, \mathrm{rad/s^2} \). This value tells us that every second, the turntable's angular velocity changes by \(2.25\) radians per second. A positive angular acceleration, as in this problem, means the object is speeding up its rotation. Conversely, a negative value would indicate slowing down.

Understanding angular acceleration is fundamental in predicting how rotational systems behave over time, especially in calculations involving changes in angular speed. It's crucial for designing mechanisms with rotating parts, like engines or gyrocompasses, where precise control of speed change is needed.

Angular Displacement

Angular displacement is a measure of the angle through which an object has rotated in a specific direction from its initial position. It's measured in radians, a unit of angular measure, which is essential in rotational motion calculations. One full rotation around a circle is equal to \(2\pi\) radians.

In our example, the turntable rotated through an angular displacement of \(60.0 \, \mathrm{rad} \) over the course of \(4.00 \, \mathrm{s} \). This displacement represents the total angle covered during this time interval. Unlike linear displacement, which is simply distance traveled, angular displacement considers how far and in what direction the rotation occurs.

• It accounts for the initial rotational position and calculates the end position in terms of the number of radians turned.
• Angular displacement can be positive or negative, indicating the direction of rotation (e.g., clockwise or counterclockwise).

Recognizing angular displacement allows us to quantify how far a rotating object has turned, forming a foundation for solving more complex rotational dynamics problems.

Initial Angular Velocity

Initial angular velocity is the speed at which an object begins to rotate, measured in radians per second (\( \mathrm{rad/s} \)). It's the starting point of angular motion, vital for understanding how the speed of rotation will change due to forces acting on the object.

In the exercise, we needed to find the initial angular velocity \( \omega_0 \) of the turntable before it experienced the given angular displacement and acceleration over \(4.00 \, \mathrm{s} \). Using the angular motion equation \( \theta = \omega_0 t + \frac{1}{2} \alpha t^2 \), the initial angular velocity can be calculated since we already have known values for the other parameters.

• Angular motion equations are similar to those used in linear motion but account for rotational contexts.
• By rearranging these formulas, we determine \( \omega_0 \) using the given time, acceleration, and displacement.

Solving for \( \omega_0 \) is crucial because it helps predict future behavior of the rotating system, such as its final speed or the distance it will cover during rotation. This concept is essential in scenarios like determining the initial spin of planets or machines.