Problem 104 - Precalculus

Question

Find the angular speed (radians/second) associated with rotating a central angle \(\theta\) in time \(t\). $$\theta=\frac{3 \pi}{4}, t=\frac{1}{6} \mathrm{sec}$$

Step-by-Step Solution

Answer

The angular speed is \( \frac{9 \pi}{2} \) radians per second.

Step 1: Identify Given Values

We are given the central angle \( \theta = \frac{3 \pi}{4} \) radians and the time \( t = \frac{1}{6} \) seconds.

Step 2: Recall Angular Speed Formula

The formula for angular speed \( \omega \) is \( \omega = \frac{\theta}{t} \). This formula relates the angular displacement to the time taken.

Step 3: Substitute Values into the Formula

Substitute the given values of \( \theta \) and \( t \) into the angular speed formula: \( \omega = \frac{\frac{3 \pi}{4}}{\frac{1}{6}} \).

Step 4: Simplify the Expression

To simplify \( \omega = \frac{\frac{3 \pi}{4}}{\frac{1}{6}} \), we multiply the numerator by the reciprocal of the denominator: \( \omega = \frac{3 \pi}{4} \times 6 \).

Step 5: Calculate Angular Speed

Simplifying further gives \( \omega = \frac{18 \pi}{4} = \frac{9 \pi}{2} \). Thus, the angular speed is \( \frac{9 \pi}{2} \) radians per second.

Key Concepts

Central Angle

A central angle is formed when two radii in a circle intersect at the circle's center. It is an important concept in circular motion. This kind of angle is directly associated with the arc that it subtends, which is a part of the circle's circumference. For our exercise, the central angle is given as \( \theta = \frac{3\pi}{4} \), which means we are dealing with a section of the circle that is measured in radians. Understanding central angles helps in calculating the movement or rotation of objects in a circle efficiently.

Angular Displacement

Angular displacement is the angle by which an object shifts on a circular path. It represents the change in the position of the object as it rotates. In our exercise, we are given an angular displacement \( \theta = \frac{3\pi}{4} \). This tells us that the object moves from its initial position an angle of \( \frac{3\pi}{4} \) radians. Unlike linear displacement, angular displacement deals with rotational motion and is measured in radians rather than units like meters.

Radians

Radians are a unit of angular measure used in many areas of mathematics. One radian corresponds to the angle created when the arc length is equal to the radius of the circle. For calculations involving rotations, radians provide a more natural way of handling angles than degrees. In the given exercise, angles are measured in radians; \( \theta = \frac{3\pi}{4} \), highlighting how radians help seamlessly integrate into formulas used for circular motion and angular speed. Radians facilitate straightforward application of mathematical relationships in circular contexts.

Mathematical Formulas

Mathematical formulas are powerful tools for solving problems like the one in our exercise. The angular speed formula, \( \omega = \frac{\theta}{t} \), is crucial for understanding circular motion. It connects the angular displacement \( \theta \) with time \( t \) to determine how quickly an object rotates. To solve the provided problem, we use this formula by substituting \( \theta \) and \( t \): \( \omega = \frac{\frac{3\pi}{4}}{\frac{1}{6}} \). Simplifying the expression gives us an exact measure of angular speed, \( \omega = \frac{9\pi}{2} \) radians per second. Mastery of these formulas allows us to predict and assess rotational movements efficiently.