Problem 178 - Calculus Volume 1

Question

A particle travels in a circular path at a constant angular speed \(\omega .\) The angular speed is modeled by the function \(\omega=9|\cos (\pi t-\pi / 12)| .\) Determine the angular speed at \(t=9 \mathrm{sec}\) .

Step-by-Step Solution

Answer

The angular speed at \( t=9 \) sec is approximately 8.69 rad/s.

Step 1: Substitute Time into the Function

To find the angular speed at a specific time, substitute the given time value into the function. The function is \( \omega = 9 |\cos (\pi t - \pi / 12)|\). We need to substitute \( t = 9 \) into this function.

Step 2: Perform the Substitution

Substitute \( t = 9 \) into the function: \[ \omega = 9 |\cos (\pi \cdot 9 - \pi / 12)| \]This simplifies to:\[ \omega = 9 |\cos (9\pi - \pi / 12)| \]

Step 3: Simplify the Expression Inside Cosine

Calculate \( \pi \cdot 9 - \pi / 12 \): First, find a common denominator for the fractions: \[ 9\pi - \pi/12 = 108\pi/12 - \pi/12 = 107\pi/12 \]

Step 4: Calculate the Cosine Value

Since cosine is periodic with period \(2\pi\), we simplify \(107\pi/12 \) to a value within a standard period by using:\[ 107\pi/12 \equiv 11\pi/12 \pmod{2\pi} \] Therefore, we need to calculate \( \cos(11\pi/12) \).

Step 5: Evaluate the Absolute Value of Cosine

\( \cos(11\pi/12) \) corresponds to \( \cos(\pi - \pi/12) \), which is equivalent to \(-\cos(\pi/12)\). Therefore:\[ |\cos(11\pi/12)| = |-\cos(\pi/12)| = \cos(\pi/12) \]

Step 6: Calculate \( \cos(\pi/12) \) Value

Using a calculator or trigonometric identity, find \( \cos(\pi/12) \). The value is approximately \(0.9659\).

Step 7: Compute the Angular Speed

Substitute \( \cos(\pi/12) \approx 0.9659\) into the function to calculate angular speed:\[ \omega = 9 \times 0.9659 \approx 8.6931 \]

Step 8: Round the Final Result

Round the angular speed to an appropriate number of decimal places, usually two decimal places in practical problems: \( \omega \approx 8.69 \) rad/s.

Key Concepts

Circular Motion

In circular motion, a particle moves along a curved path, always at a fixed distance from a central point. This path is often a circle with the particle experiencing a constant angular speed, denoted by \( \omega \), which represents how quickly it rotates around a central axis.
Angular speed differs from linear speed, which is based on the distance traveled over time. Instead, angular speed focuses on the angle covered during motion. In this problem, the particle's angular speed is expressed as a function \( \omega=9|\cos(\pi t - \pi/12)| \). Understanding this function helps us predict the particle's speed at any time \( t \).
To solve for the angular speed at a particular moment (like \( t=9 \) seconds), you need to substitute the time into the equation and calculate its value. Circular motion can be mesmerizing, thanks to its consistency and predictability, and grasping this concept is essential for understanding how objects move in circular paths, from planets orbiting stars to wheels spinning on an axis.

Trigonometric Functions

Trigonometric functions play a vital role in describing and modeling circular motion. The function used in this exercise, \( \omega=9|\cos(\pi t - \pi/12)| \), is built on the cosine function, which is a core part of trigonometry.
The cosine function helps to relate an angle of rotation to the x-coordinate in the unit circle, or more simply, the horizontal distance of a point from the circle's center. The absolute value ensures that the angular speed remains non-negative, reflecting the idea that speed cannot be negative, only direction.
By plugging in the value of \( t=9 \), the function undergoes transformations that involve calculating cosine of an angle, in this case \( \cos(11\pi/12) \). Understanding how trigonometric functions like cosine work provides insights into periodic behaviors in various applications, such as waves and oscillations.

Periodic Functions

Periodic functions recur regularly, repeating their values at fixed intervals. In this exercise, \( \omega=9|\cos(\pi t - \pi/12)| \) represents a periodic function due to its reliance on the cosine function, which is naturally periodic with a period of \( 2\pi \).
This periodicity means that the angular speed pattern repeats and can be easily predicted over time. The function's periodicity allows you to simplify complex expressions by reducing angles larger than \( 2\pi \) to an equivalent angle (e.g., \( 107\pi/12 \) to \( 11\pi/12 \)).
Understanding periodic functions is key to mastering concepts related to cycles and rhythms that appear in numerous natural and engineered systems, like signal processing and seasonal patterns.