Problem 63 - Algebra 2

Question

Angular velocity is defined by the equation \(\omega=\frac{\theta}{t},\) where \(\theta\) is usually expressed in radians and \(t\) represents time. Find the angular velocity in radians per second of a point on a bicycle tire if it completes 2 revolutions in 3 seconds. F. \(\frac{\pi}{3}\) G. \(\frac{\pi}{2}\) H. \(\frac{2 \pi}{3}\) J. \(\frac{4 \pi}{3}\)

Step-by-Step Solution

Answer

The angular velocity is \(\frac{4\pi}{3}\) radians per second, option J.

Step 1: Understand the Problem

We are given the number of revolutions a bicycle tire completes in a certain period of time and asked to find the angular velocity.

Step 2: Convert Revolutions to Radians

Since 1 revolution corresponds to an angle of \(2\pi\) radians, 2 revolutions will be \(2 \times 2\pi = 4\pi\) radians.

Step 3: Identify Given and Known Values

We know \(\theta = 4\pi\) radians and \(t = 3\) seconds from the problem.

Step 4: Apply the Angular Velocity Formula

The formula for angular velocity is \(\omega = \frac{\theta}{t}\). Substitute the known values into the formula: \(\omega = \frac{4\pi}{3}\).

Step 5: Match Solution to Provided Answer

Compare the calculated angular velocity \(\frac{4\pi}{3}\) with the provided options. It corresponds to option J.

Key Concepts

Radians

Radians are a unit of measure used in angular motion. In simpler terms, a radian represents the angle created when the radius of a circle is wrapped along its circumference. This makes understanding angles more intuitive when dealing with circles and circular motion.

• One full circle is equal to 360 degrees, but in radians, it is represented as \(2\pi\).
• This means that \(\pi\) radians equal 180 degrees.

In the context of the given problem, radians are crucial because we describe the angle through which the bicycle tire rotates in terms of \(\pi\). This conversion from revolutions to radians is important for accurately calculating angular velocity.

Revolutions

Revolutions are another way of describing the motion of an object around a circular path. A revolution refers to a complete 360-degree turn around a circle. Understanding how to convert revolutions to radians is important when working with circular motion problems.

• One revolution is equal to \(2\pi\) radians, which is equivalent to the angle around a full circle.
• In calculations, we often convert this quantity to radians, because it simplifies formulas and mathematical expressions related to circles.

In our example, the problem states the bicycle tire completes 2 revolutions. By converting 2 revolutions to radians, \(2 \times 2\pi = 4\pi\), we can use this information to find the tire's angular velocity.

Angular Motion

Angular motion refers to how an object moves through a circular path, just like our bicycle wheel spinning. It involves various aspects, such as angular velocity, which is the rate of change of angular displacement over time.

• Angular velocity, denoted by \(\omega\), measures how fast something spins or rotates.
• It is determined using the formula \(\omega = \frac{\theta}{t}\), where \(\theta\) is the angle (in radians) and \(t\) is the time.

The problem demonstrates how to determine angular velocity by using both the angle in radians and the time taken. Knowing how to handle angular motion helps us understand rotating objects in our everyday life, from wheels to spinning tops.

Time

Time plays a crucial role in calculating angular velocity as it provides the period over which a circular motion occurs. In physics, time is typically expressed in seconds.

• Time enables us to measure how fast or slow a motion is occurring.
• Time, when combined with angle measurements, directly influences the computation of angular velocity.

In the bicycle wheel scenario, it took 3 seconds for the wheel to make 2 revolutions. This information, along with the converted angle in radians, allowed for the application of the angular velocity formula \(\omega = \frac{\theta}{t}\), resulting in the calculation \(\omega = \frac{4\pi}{3}\). Without the time component, determining the speed of rotation would be impossible.