Problem 175 - Calculus Volume 1

Question

As a point P moves around a circle, the measure of the angle changes. The measure of how fast the angle is changing is called angular speed, \(\omega,\) and is given by \(\omega=\theta / t, \quad\) where \(\theta\) is in radians and \(t\) is time. Find the a. \(\theta=\frac{7 \pi}{4} \mathrm{rad}, t=10 \mathrm{sec} \quad\) b. \(\theta=\frac{3 \pi}{5} \mathrm{rad}, t=8 \quad \mathrm{sec} \quad\) c. \(\theta=\frac{2 \pi}{9} \mathrm{rad}, t=1 \mathrm{min} \quad\) d. \(\theta=23.76 \mathrm{rad}, t=14 \mathrm{min}\)

Step-by-Step Solution

Answer

Angular speeds are \( \frac{7 \pi}{40} \), \( \frac{3 \pi}{40} \), \( \frac{\pi}{270} \), and \( 0.0283 \) rad/sec.

Step 1: Understanding Angular Speed Formula

To find angular speed \( \omega \), use the formula \( \omega = \frac{\theta}{t} \) where \( \theta \) is the angle in radians, and \( t \) is the time interval.

Step 2: Calculate Angular Speed for A

Given \( \theta = \frac{7 \pi}{4} \) rad and \( t = 10 \) sec, apply the formula: \( \omega = \frac{\frac{7 \pi}{4}}{10} = \frac{7 \pi}{40} \) rad/sec.

Step 3: Calculate Angular Speed for B

Given \( \theta = \frac{3 \pi}{5} \) rad and \( t = 8 \) sec, apply the formula: \( \omega = \frac{\frac{3 \pi}{5}}{8} = \frac{3 \pi}{40} \) rad/sec.

Step 4: Convert Units for C

Time \( t = 1 \) min must be converted to seconds: \( 1 \) min = \( 60 \) sec. Then apply the formula for \( \theta = \frac{2 \pi}{9} \) rad: \( \omega = \frac{\frac{2 \pi}{9}}{60} = \frac{2 \pi}{540} = \frac{\pi}{270} \) rad/sec.

Step 5: Convert Units for D

Time \( t = 14 \) min must be converted to seconds: \( 14 \) min = \( 840 \) sec. For \( \theta = 23.76 \) rad, \( \omega = \frac{23.76}{840} = 0.0282857 \approx 0.0283 \) rad/sec.

Key Concepts

radian

Understanding the concept of a radian is essential when dealing with angular movement. A radian is a standard unit of angular measure used in many areas of math. It describes the angle made when the radius of a circle is wrapped along its circumference.
One radian is the angle created at the center of a circle by an arc whose length is equal to the circle's radius.
This works out to about 57.3 degrees, but in calculations, radians offer a more natural and precise way to handle angle measurement.
You'll find radians convenient particularly because they lead to simpler formulas in calculus and trigonometry.

• A circle has a total of 2π radians, as the circumference is 2π times the radius.
• The conversion between radians and degrees is straightforward: Multiply by 180/π to convert from radians to degrees.

angular velocity

Angular velocity, often referred to using the Greek letter omega (\( \omega \)), quantifies how quickly an object rotates or revolves relative to another point, typically the center of a circle. It is a vector quantity, having both a magnitude and a direction.
The magnitude of angular velocity (\( \omega \)) is measured in radians per second.
Mathematically, it's expressed as:
\[\omega = \frac{\theta}{t}\]where \( \theta \) is the angular displacement in radians, and \( t \) is the time taken for this displacement.
Understanding angular velocity is crucial in fields where rotational motion is key, such as astronomy, mechanical engineering, and robotics.

• Example: If a point on the edge of a wheel travels through an angle of \( \pi \) radians in 2 seconds, its angular velocity is \( \frac{\pi}{2} \) rad/s.
• Angular velocity can tell us how fast a rotating system is turning without needing to consider the object's size.

conversion of units

Converting units appropriately is an important step when calculating angular speed. Since the formula for angular velocity uses radians and time in seconds, ensuring consistency in units is key.
The most common conversion you might need is time, particularly from minutes to seconds.
Remember, 1 minute equals 60 seconds.
Changing between units often requires conversion factors, and it's always a good practice to double-check your units before proceeding with calculations.

• When dealing with extended time periods, always convert to seconds for \( \omega \) calculations.
• Example: In the provided exercise, 14 minutes converts to \( 840 \) seconds by multiplying by \( 60 \).
• This ensures accuracy and conformity with the standard units used in physics and mathematics.