Problem 75 - Algebra and Trigonometry

Question

In Exercises \(75-76,\) express each angular speed in radians per second. 6 revolutions per second

Step-by-Step Solution

Answer

The angular speed is \(12\pi\) radians per second.

Step 1: Identify given angular speed

The given angular speed is 6 revolutions per second.

Step 2: Apply conversion factor

Next, apply the conversion factor from revolutions to radians which is \(2\pi\) radians per revolution. Multiplying 6 revolutions per second by this factor gives \(6*2\pi\) radians per second.

Step 3: Simplify the expression

Finally, simplify the expression to give the angular speed in radians per second. The solution is \(12\pi\) radians per second.

Key Concepts

Understanding 'Radians per Second'

A radian is a unit of angular measure used in many areas of mathematics. When we talk about angular speed in terms of 'radians per second', it refers to how much angle is swept per unit of time. Unlike degrees, which measures angles in parts out of 360, radians measure angles based on the circumference of a circle. A full circle is equivalent to 2π radians.

If you imagine a circle with a point moving along its edge, the number of radians per second tells you how fast this point is moving in terms of angle. It defines the rotational speed and allows us to work with linear measurements that fit nicely into mathematical calculations.

• Radians are a central concept in trigonometry and calculus.
• They simplify many formulas, reducing the possible chance of errors in calculations.

Grasping the concept of radians per second is crucial for solving problems involving circular motion.

Revolutions per Second Described

'Revolutions per second' is another way to measure angular speed. It indicates how many complete turns or circles are being completed in one second. A revolution can be thought of as completing one full turn around a circle, equating to 360 degrees or 2π radians.

In contexts of motors, wheels, and many engineering applications, you often use revolutions per second or per minute to describe the speed of rotational objects. This makes it an important unit in physics and engineering.

• Helps describe circular motion in mechanical systems.
• Commonly measured in revolutions per minute (RPM) for ease of understanding in many applications.

Understanding revolutions per second provides an intuitive grasp of how fast something is spinning.

What is the Conversion Factor?

The conversion factor allows you to switch between different units of measurement. In terms of angular speed, there are mainly two units of measure: radians and revolutions. The conversion factor between these two is 2π, because there are 2π radians in one complete revolution of a circle.

Utilizing the conversion factor enables us to convert an angular speed given in revolutions per second to radians per second.

• Use the equation: Angular speed in radians/s = Angular speed in revolutions/s × 2π radians.
• Simplifies conversion and increases accuracy in calculations.

Employing conversion factors effectively bridges different measurement systems, making the math easier and more consistent for engineers and scientists.

Mathematical Expression Simplification

Simplifying mathematical expressions is about making them easier to handle and interpret. When we simplify, we aim to transform the expression into its most reduced form without changing its value. Consider our problem where we convert 6 revolutions per second to radians per second.

After applying the conversion factor, we arrive at the expression 6 * 2π. Simplifying this multiplication results in 12π radians per second.

• Simplifying expressions ensures clarity and precision.
• Leads to more efficient and comprehensible results.

This simplification process is essential in mathematical calculations, particularly when dealing with complex formulas or large datasets. It helps avoid numerical errors and confusion, ultimately leading to better understanding and application of mathematical principles.