Problem 136 - Precalculus

Question

In calculus we work with real numbers; thus, the measure of an angle must be in radians. An object is rotating at \(600^{\circ}\) per second, find the central angle \(\theta,\) in radians, when \(t=3 \mathrm{sec}\).

Step-by-Step Solution

Answer

The central angle \(\theta\) is \(10\pi\) radians after 3 seconds.

Step 1: Convert Degrees to Radians

The first step is to convert the angular speed from degrees per second to radians per second. We use the formula: \( 1^{\circ} = \frac{\pi}{180} \text{ radians} \). Therefore, \(600^{\circ}/\text{sec} = 600 \times \frac{\pi}{180} \text{ radians/second} = \frac{10\pi}{3} \text{ radians/second} \).

Step 2: Calculate the Central Angle After 3 Seconds

To find the central angle \( \theta \) in radians after 3 seconds, we need to multiply the angular velocity by time. This gives: \( \theta = \left( \frac{10\pi}{3} \text{ radians/second} \right) \times 3 \text{ seconds} = 10\pi \text{ radians} \).

Key Concepts

Angular Velocity

Angular velocity is a measure of how quickly an object rotates or revolves relative to another point. This is usually expressed in terms of radians per second in scientific calculations, especially when dealing with calculus-related problems.

Angular velocity highlights the rotational speed of an object, not to be confused with linear velocity, which refers to straight-line motion. Here, angular velocity is essentially the angle swept by the object per unit of time. This becomes particularly relevant when we analyze systems like spinning wheels or celestial bodies.

For example, in our exercise, the object was spinning at a speed of \(600^{\circ} \text{ per second}\). Converting this to radians per second is crucial because radians are the standard unit for angle measurements in mathematical analysis. Ensuring angular velocity is in radians allows for a straightforward application of the formulas used in calculus and physics.

Angle Conversion

Most angles can be expressed in degrees or radians. The conversion between these units is central in many mathematical and physical applications. Commonly, angles are given in degrees, while radians are more convenient for integration or deriving other formulas due to their natural relation to the circle's arc length.

To convert from degrees to radians, use the conversion factor \( \frac{\pi}{180} \). This means \(1^{\circ}\) is equivalent to \(\frac{\pi}{180}\) radians. So, when converting \(600^{\circ}\) per second in the exercise, we multiply by \(\frac{\pi}{180}\) to obtain the angular velocity in radians per second.

Why is this conversion important? Radians provide a direct measurement proportional to the radius of the circle, thus simplifying problems related to arc lengths and areas. It helps in maintaining consistency across accumulated measures and making calculus terms cleaner.

Central Angle

The central angle is the angle that forms when two radii of a circle meet at the circle's center. It provides a useful way to describe circular motion and is often represented in radians.

In the context of rotational motion, such as the given exercise, the central angle helps quantify the extent of rotation over a period of time. For instance, if you know the angular velocity and the time duration, you can calculate the central angle as occurs in our step-by-step solution.

Calculating the central angle involves multiplying the angular velocity by time. This gives a measure of how much of the circumference the object has traced. So, with \(600^{\circ}\) per second converted to \(\frac{10\pi}{3} \text{ radians/second}\), multiplying by 3 seconds, we deduce the central angle \(\theta = 10\pi\) radians. This demonstrates how the object completes multiple full rotations, connecting back to understanding radians are related to complete circle rotations, or \(2\pi\) radians per revolution.