Problem 103 - Precalculus

Question

Find the angular speed (radians/second) associated with rotating a central angle \(\theta\) in time \(t\). $$\theta=25 \pi, t=10 \mathrm{sec}$$

Step-by-Step Solution

Answer

The angular speed is \( \frac{5\pi}{2} \) radians/second.

Step 1: Understand the Relationship

Given that the angular speed (c9) is defined as the angle turned per unit of time, we can write the formula for angular speed as \( \omega = \frac{\theta}{t} \). In this formula, \( \theta \) represents the angle in radians, and \( t \) is the time in seconds.

Step 2: Substitute the Given Values

Substitute the given values \( \theta = 25\pi \) and \( t = 10 \) seconds into the formula \( \omega = \frac{\theta}{t} \). This gives us: \( \omega = \frac{25\pi}{10} \).

Step 3: Simplify the Equation

Simplify the fraction \( \frac{25\pi}{10} \) by dividing both the numerator and the denominator by their greatest common divisor, which is 5. This leads to \( \omega = \frac{5\pi}{2} \).

Step 4: Conclude with Units of Measurement

The resulting angular speed is \( \omega = \frac{5\pi}{2} \) radians per second. Ensure your answer states the units of measurement clearly, as the problem requires angular speed in radians per second.

Key Concepts

Radians

Radians are a unit of angular measure. They offer a way of measuring angles based on the radius of a circle. Unlike degrees, which divide a circle into 360 parts, a radian measures the length of the arc created by the angle, using the radius as a reference. This means:

• One radian is the angle created when the arc length is equal to the radius of the circle.
• A full circle, therefore, contains \( 2\pi \) radians because the circumference of a circle is \( 2\pi r \), where \( r \) is the radius.
• This makes conversion between degrees and radians straightforward: \( 180^\circ = \pi \) radians.

Understanding radians is crucial when discussing angular speed because they efficiently translate rotational motion into linear terms. This efficiency stems from the relationship between the circle's radius and arc length.

Central Angle

A central angle is the angle formed by two radii in a circle. It opens from the center and points outwards, slicing a segment of the circle. The size of this angle in radians directly correlates with the arc length. Key points about central angles include:

• When expressed in radians, the central angle \( \theta \) relates to the arc length "\( l \)" and the radius "\( r \)" of a circle by the formula \( \theta = \frac{l}{r} \).
• Central angles allow us to calculate angular speed when the angle swept over time is known.
• In the context of a rotating object, knowing the central angle helps determine how far and in what direction an object has rotated.

In our exercise, understanding the central angle \( \theta = 25\pi \) allows us to calculate how fast it moves, which is essential in determining angular speed.

Angular Velocity

Angular velocity, also known as angular speed, is a concept used to describe the speed of rotation around a circular path. It's an important measure in circular motion:

• Angular velocity is denoted by \( \omega \) and is typically measured in radians per second.
• The formula \( \omega = \frac{\theta}{t} \) provides a way to calculate angular velocity given a central angle \( \theta \) in radians and a time interval \( t \).
• A higher angular velocity means an object is rotating faster.

Understanding angular velocity is crucial in scenarios involving rotational movement, such as the rotation of wheels or gears. In our example, we calculated the angular velocity based on an angle of \( 25\pi \) radians and a time of 10 seconds, resulting in \( \omega = \frac{5\pi}{2} \) radians per second. This helps in determining how quickly an object is rotating per unit of time.