Question

A propeller on a ship has an initial angular velocity of \(5.1 \mathrm{rad} / \mathrm{s}\) and an angular acceleration of \(1.6 \mathrm{rad} / \mathrm{s}^{2}\). What is the angular velocity of the propeller after \(3.0 \mathrm{~s}\) ?

Short answer

The angular velocity after 3.0 seconds is 9.9 rad/s.

Step-by-step solution

Identify Given Values

The problem provides the initial angular velocity \( \omega_0 = 5.1 \ \text{rad/s} \) and the angular acceleration \( \alpha = 1.6 \ \text{rad/s}^2 \). It also provides the time duration \( t = 3.0 \ \text{s} \).

Use the Angular Velocity Formula

The formula to find the final angular velocity \( \omega \) is given by \( \omega = \omega_0 + \alpha t \). This equation relates initial velocity, angular acceleration, and time to find the final velocity.

Substitute Given Values into the Formula

Substitute the given values into the equation: \( \omega = 5.1 \ \text{rad/s} + (1.6 \ \text{rad/s}^2)(3.0 \ \text{s}) \).

Calculate the Result

After performing the multiplication, we have \( 1.6 \times 3.0 = 4.8 \). Therefore, substitute back to get \( \omega = 5.1 + 4.8 = 9.9 \ \text{rad/s} \).

Key concepts

Angular Acceleration

Angular acceleration is a measure of how quickly the angular velocity of an object is changing over time. It's similar to linear acceleration, but it applies to objects that are rotating rather than moving in a straight line. If you think about spinning a wheel, angular acceleration is how fast you're speeding up or slowing down that spin. - **Unit:** Angular acceleration is typically measured in radians per second squared \(( \mathrm{rad/s}^2 )\). - **Direction:** It can be positive (object speeds up) or negative (object slows down).In our example with the ship's propeller, the angular acceleration was given as \(1.6 \ \mathrm{rad/s}^2\). This means that every second, the angular velocity of the propeller increases by \(1.6 \ \mathrm{rad/s}\). So, over three seconds, the total increase in angular velocity is \(1.6 \times 3 = 4.8 \ \mathrm{rad/s}\). Angular acceleration is a crucial aspect in understanding rotational dynamics because it tells us how forces are influencing the speed of rotation.

Kinematics

Kinematics is a branch of physics that deals with the motion of objects without considering the forces that cause these motions. In the context of rotational motion, kinematics is concerned with quantities like angular displacement, angular velocity, and angular acceleration. - **Angular Variables:** When studying rotations, we replace linear concepts with their angular counterparts, such as: - Displacement with angular displacement \( \theta \) - Velocity with angular velocity \( \omega \) - Acceleration with angular acceleration \( \alpha \)In the given exercise, kinematics provides the tools to predict how the propeller's rotational motion will change over time, using the initial conditions and constants, such as initial angular velocity and angular acceleration. By applying the formula \( \omega = \omega_0 + \alpha t \), we directly use kinematical relationships to find out how the angular velocity progresses as time advances.Kinematics allows us to predict future motion based on initial velocities and accelerations, crucial for problem-solving in physics contexts. Understanding this helps one to model and anticipate various physical phenomena effectively.

Rotational Motion

Rotational motion occurs when an object spins around an internal axis. Unlike linear motion, where objects move in a straight line, rotational motion deals with the movement of objects in a circular path.- **Key Quantities:** In rotational motion, the primary quantities are: - **Angular Displacement (\theta):** The angle through which an object has rotated in a given time. - **Angular Velocity (\omega):** The rate of change of angular displacement, or how fast something is rotating. - **Angular Acceleration (\alpha):** The rate of change of angular velocity.The ship propeller example illustrates how rotational motion works in real life. You observe a propeller starting at a certain angular velocity, being influenced by an angular acceleration, and ultimately increasing its angular velocity over time. Through formulas and calculations, we can precisely define the state of rotational motion at any point in time, like determining that after \(3.0 \, \mathrm{s} \) the angular velocity of our propeller is \(9.9 \, \mathrm{rad/s}\). Understanding rotational motion is crucial for navigating scenarios involving wheels, gears, turbines, and other rotating systems, as it lays the foundation for designing and analyzing such systems efficiently.