Question

An object has an angular velocity of $1.0\mathrm{rad}/\mathrm{s}$ and an angular acceleration of $-0.5\mathrm{rad}/{\mathrm{s}}^2$. Is the speed of its rotation increasing or decreasing?

Short answer

The rotation speed is decreasing because the negative angular acceleration opposes the angular velocity.

Step-by-step solution

Understand the Concepts

To determine if the rotational speed is increasing or decreasing, we need to evaluate the angular velocity and angular acceleration. Angular velocity reflects how fast an object is rotating, while angular acceleration describes how the angular velocity is changing over time. A negative angular acceleration indicates that the angular velocity is decreasing if it is in the same direction as angular velocity.

Identify Given Values

Given in the problem: - Angular velocity, $\omega =1.0\text{rad/s}$- Angular acceleration, $\alpha =-0.5{\text{rad/s}}^2$

Analyze the Direction

Since the angular velocity $\omega =1.0\text{rad/s}$ is positive, the object is rotating in a positive direction. However, the angular acceleration $\alpha =-0.5{\text{rad/s}}^2$ is negative, indicating that it is acting in the opposite direction to the angular velocity.

Conclusion Based on Angular Acceleration

Since the angular acceleration is negative and opposite to the direction of angular velocity, it implies that the angular velocity is decreasing over time. Therefore, the speed of the object's rotation is decreasing.

Key concepts

Angular Velocity

Angular velocity is a key concept in rotational motion, functioning quite similarly to linear velocity in linear motion. It represents the rate at which an object rotates about an axis. Simply put, it tells us how fast something is spinning.

- **Units:** Angular velocity is measured in radians per second ( $\mathrm{rad}/\mathrm{s}$ ).
- **Direction:** Just as with vectors, angular velocity has both a magnitude and a direction. The direction is usually specified by the right-hand rule, where the thumb points along the axis of rotation, and your fingers curl in the direction of rotation.

A positive angular velocity indicates clockwise rotation, while a negative value suggests counter-clockwise rotation. Understanding this concept helps you analyze how rotational speed changes over time, whether the object is spinning faster or slower.

Angular Acceleration

Angular acceleration describes how the angular velocity changes with time. It is a vital concept in understanding whether an object is spinning up or slowing down.

- **Units:** It is measured in radians per second squared ( $\mathrm{rad}/{\mathrm{s}}^2$ ).
- **Positive or Negative:** Much like in linear motion with acceleration, the sign of angular acceleration matters.

- A positive angular acceleration means that the angular velocity is increasing. In other words, the object is rotating faster.
- Conversely, a negative value indicates that the angular velocity is decreasing, meaning the object is slowing down in its rotation.

When analyzing rotational motion, it's vital to compare the signs of angular velocity and angular acceleration. If they have opposite signs, the spinning will slow down, as is the case in the given exercise. The angular acceleration acts against the angular velocity, reducing it over time.

Dynamics of Rotation

The dynamics of rotation is akin to understanding how forces cause motion in linear kinetics, but now applied to rotational movements. It deals with the factors affecting an object’s rotation such as torque, moment of inertia, and angular velocity.

- **Torque:** This is the rotational equivalent of force. It measures how effectively a force causes an object to rotate. Torque can be calculated by multiplying the force applied and the distance from the axis of rotation ( $Torque=Force\times Distance$ from axis).
- **Moment of Inertia:** This tells us how difficult it is to change an object’s current state of rotation. It depends on the mass of the object and its distribution relative to the axis of rotation. The more spread out the mass is, the greater the inertia.
- **Newton's Second Law for Rotation:** This is expressed as $Torque=I\cdot \alpha$ , where $I$ is the moment of inertia and $\alpha$ is the angular acceleration.

These components together dictate whether an object’s rotation will speed up or slow down, similar to how force, mass, and acceleration interplay in linear dynamics. Understanding these basics helps in analyzing and predicting rotational behavior comprehensively.