Problem 18 - Technical Mathematics with Calculus

Question

A link in a mechanism rotating with an angular velocity of \(3.00 \mathrm{rad} / \mathrm{s}\) is given an acceleration of \(5.00 \mathrm{rad} / \mathrm{s}^{2}\) at \(t=0 .\) Find the angular velocity after \(20.0 \mathrm{s}\)

Step-by-Step Solution

Answer

\(\omega = 103.00 \,\mathrm{rad/s}\)

Step 1: Identify the Known Quantities

We are given the initial angular velocity, \(\omega_0 = 3.00 \,\mathrm{rad/s}\), and the angular acceleration, \(\alpha = 5.00\, \mathrm{rad/s^2}\). The time interval is \(t = 20.0 \,\mathrm{s}\).

Step 2: Use the Angular Velocity Formula

To find the angular velocity after a certain time when angular acceleration is constant, use the equation \(\omega = \omega_0 + \alpha \cdot t\), where \(\omega\) is the final angular velocity.

Step 3: Substitute Known Values into the Equation

Substitute the known quantities into the equation to find the final angular velocity: \(\omega = 3.00 \,\mathrm{rad/s} + (5.00 \,\mathrm{rad/s^2})(20.0 \,\mathrm{s})\).

Step 4: Calculate the Final Angular Velocity

Perform the multiplication and addition to find \(\omega\): \(\omega = 3.00 \,\mathrm{rad/s} + 100.00 \,\mathrm{rad/s}\), which simplifies to \(\omega = 103.00 \,\mathrm{rad/s}\).

Key Concepts

Angular Acceleration

Angular acceleration is central to understanding how objects in rotational motion change their speed. It's the rate at which an object's angular velocity changes. In the context of the exercise from the textbook, we've explored an example where a link in a rotating mechanism is accelerated at a rate of \(5.00 \, \mathrm{rad/s^2}\) starting from an initial velocity of \(3.00 \, \mathrm{rad/s}\).

This concept is analogous to linear acceleration but for rotational motion; it's just applied to something that spins instead of something that moves in a straight line. A positive angular acceleration means an object is speeding up, while a negative one indicates it's slowing down. In the given problem, since the value of angular acceleration is positive, the link's rotational speed increases over time.

Kinematics in Physics

Kinematics is the branch of physics that describes the motion of points, bodies (objects), and systems of bodies (groups of objects) without considering the forces that cause them to move. Essentially, it focuses on displacement, velocity, and acceleration, and in the case of rotational motion, these translate to angular displacement, angular velocity, and angular acceleration.

Kinematic equations provide simple relationships between these parameters, allowing us to predict the future position or state of motion of an object. For rotational motion, we have a similar set of equations to those in linear kinematics. The simple formula \(\omega = \omega_0 + \alpha \cdot t\), used in the exercise solution, is part of the kinematic formulas and represents a powerful tool for predicting an object's angular velocity when its angular acceleration is constant.

Rotational Motion

Rotational motion pertains to objects that turn around a fixed axis. Examples include a merry-go-round, a spinning top, and Earth's rotation about its axis. In the textbook exercise, rotational motion is exemplified by a link in a mechanism that rotates.

Understanding rotational motion is crucial in various applications: from simple machinery to complex space physics. One of the key aspects of rotational motion is that all points in the body have the same angular velocity and angular acceleration, but their linear velocities and accelerations differ depending on their distance from the axis of rotation. This concept helps explain why the outer edge of a spinning disk moves faster than a point closer to the center, even though both complete a full rotation in the same amount of time.

It’s important to grasp these concepts to understand how objects rotate and how their rotational motion can be described quantitatively using measurements such as radians per second for angular velocity, and radians per second squared for angular acceleration.