91Ó°ÊÓ

Express each of the following in terms of other angular measures: (a) \(28^{\circ}\), (b) \(\frac{1}{4} \mathrm{rev} / \mathrm{s}\), (c) \(2.18 \mathrm{rad} / \mathrm{s}^{2}\). (a) \(28^{\circ}=(28\) deg \()\left(\frac{1 \mathrm{rev}}{360 \mathrm{deg}}\right)=0.078 \mathrm{rev}\) \(=(28 \mathrm{deg})\left(\frac{2 \pi \mathrm{rad}}{360 \mathrm{deg}}\right)=0.49 \mathrm{rad}\) (b) \(\begin{aligned} \frac{1}{4} \frac{\mathrm{rev}}{\mathrm{s}} &=\left(0.25 \frac{\mathrm{rev}}{\mathrm{s}}\right)\left(\frac{360 \mathrm{deg}}{1 \mathrm{rev}}\right)=90 \frac{\mathrm{deg}}{\mathrm{s}} \\ &=\left(0.25 \frac{\mathrm{rev}}{\mathrm{s}}\right)\left(\frac{2 \pi \mathrm{rad}}{1 \mathrm{rev}}\right)=\frac{\pi}{2} \frac{\mathrm{rad}}{\mathrm{s}} \end{aligned}\) (c) \(\begin{aligned} 2.18 \frac{\mathrm{rad}}{\mathrm{s}^{2}} &=\left(2.18 \frac{\mathrm{rad}}{\mathrm{s}^{2}}\right)\left(\frac{360 \mathrm{deg}}{2 \pi \mathrm{rad}}\right)=125 \frac{\mathrm{deg}}{\mathrm{s}^{2}} \\ &=\left(2.18 \frac{\mathrm{rad}}{\mathrm{s}^{2}}\right)\left(\frac{1 \mathrm{rev}}{2 \pi \mathrm{rad}}\right)=0.347 \frac{\mathrm{rev}}{\mathrm{s}^{2}} \end{aligned}\)

Step by step solution

01

Convert Degrees to Revolutions

Convert \(28^{\circ}\) to revolutions using the conversion \(1 \text{ rev} = 360^{\circ}\):\[28^{\circ} = 28 \left( \frac{1 \text{ rev}}{360 \text{ deg}} \right) = 0.078 \text{ rev}\]

02

Convert Degrees to Radians

Convert \(28^{\circ}\) to radians using the conversion \(2\pi \text{ rad} = 360^{\circ}\):\[28^{\circ} = 28 \left( \frac{2\pi \text{ rad}}{360 \text{ deg}} \right) = 0.49 \text{ rad}\]

03

Convert Revolutions per Second to Degrees per Second

Convert \(\frac{1}{4} \text{ rev/s}\) to degrees per second:\[\frac{1}{4} \text{ rev/s} = 0.25 \text{ rev/s} \times \frac{360 \text{ deg}}{\text{ rev}} = 90 \text{ deg/s}\]

04

Convert Revolutions per Second to Radians per Second

Convert \(\frac{1}{4} \text{ rev/s}\) to radians per second:\[\frac{1}{4} \text{ rev/s} = 0.25 \text{ rev/s} \times \frac{2\pi \text{ rad}}{\text{ rev}} = \frac{\pi}{2} \text{ rad/s}\]

05

Convert Radians per Second Squared to Degrees per Second Squared

Convert \(2.18 \text{ rad/s}^2\) to degrees per second squared:\[2.18 \text{ rad/s}^2 = 2.18 \times \frac{360 \text{ deg}}{2\pi \text{ rad}} = 125 \text{ deg/s}^2\]

06

Convert Radians per Second Squared to Revolutions per Second Squared

Convert \(2.18 \text{ rad/s}^2\) to revolutions per second squared:\[2.18 \text{ rad/s}^2 = 2.18 \times \frac{1 \text{ rev}}{2\pi \text{ rad}} = 0.347 \text{ rev/s}^2\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Degrees to Radians

Understanding how to convert degrees to radians is crucial in angular measurement. This conversion is commonly used in mathematics and physics, as it allows for easier calculation in trigonometry and calculus. Degrees measure angles with the full circle being 360 degrees. Radians, on the other hand, measure angles based on the radius of a circle. In a complete circle, there are \(2\pi\) radians. To convert degrees to radians, you multiply the number of degrees by \(\frac{\pi}{180}\). For example, to convert \(28^{\circ}\) to radians:\[28^{\circ} \times \frac{\pi}{180^{\circ}} = 0.49 \, \text{rad}\]This conversion formula is vital for many applications in science and engineering, where radian measure is the standard.

Revolutions per Second

Revolutions per second (rev/s) is a unit used to express angular speed, which tells how fast an object is rotating. Understanding this concept is essential in fields like mechanical engineering and physics. One revolution is a complete turn around a circle, equivalent to \(360^{\circ}\) or \(2\pi\) radians. When mentioned as rev/s, it measures how many full circles an object completes in one second.To convert revolutions per second to degrees per second or radians per second, simply use the equivalency of 1 rev = \(360^{\circ}\) or 1 rev = \(2\pi \) rad.For example, converting \(\frac{1}{4} \, \text{rev/s}\) to degrees per second:\[0.25 \, \text{rev/s} \times 360 \, \text{deg/rev} = 90 \, \text{deg/s}\]And to radians per second:\[0.25 \, \text{rev/s} \times 2\pi \, \text{rad/rev} = \frac{\pi}{2} \, \text{rad/s}\]

Radians per Second Squared

Radians per second squared (rad/s²) is the unit of angular acceleration. It describes how quickly the angular velocity of an object is changing over time. This measure is crucial in understanding rotating systems, especially in physics and engineering.Angular acceleration, in radians per second squared, tells us how the speed of rotation is increasing or decreasing. To translate this into other units like degrees per second squared or revolutions per second squared, use the basic conversions:1 rad/s² equals \( \frac{180}{\pi} \) deg/s².Additionally, 1 rad/s² equals \( \frac{1}{2\pi} \) rev/s².For instance, converting \(2.18 \, \text{rad/s}^2\) into degrees per second squared:\[2.18 \, \text{rad/s}^2 \times \frac{180}{\pi} \, \text{deg/rad} = 125 \, \text{deg/s}^2\]And into revolutions per second squared:\[2.18 \, \text{rad/s}^2 \times \frac{1}{2\pi} \, \text{rev/rad} = 0.347 \, \text{rev/s}^2\]

Angular Velocity

Angular velocity is a vector quantity that represents the rate of rotation about an axis. It provides us with information on how fast an object rotates or spins. This is essential in various applications, from satellite motion analysis to simple mechanical clocks.Typically measured in radians per second (rad/s) or degrees per second (deg/s), angular velocity indicates the change of angular position of an object with time.For example, a wheel spinning at \(90 \, \text{deg/s}\) indicates that it rotates 90 degrees every second. In radians, the same motion equates to:\[90 \, \text{deg/s} \times \frac{\pi}{180 \, \text{deg}} = \frac{\pi}{2} \, \text{rad/s}\]By understanding angular velocity, you can predict where an object will be at a future time, assuming constant speed.

Angular Acceleration

Angular acceleration is the rate of change of angular velocity over time. It's a measure of how quickly an object speeds up or slows down its rotation.Just like linear acceleration measures changes in linear speed, angular acceleration measures changes in rotational speed. Common units for this are radians per second squared (rad/s²) or degrees per second squared (deg/s²).In practice, understanding angular acceleration helps with designing systems that involve rotational dynamics, such as car engines or wind turbines. For example, if an object accelerates from \(0\) to \( \frac{\pi}{2} \, \text{rad/s} \) in \(5\) seconds, its angular acceleration is calculated as:\[ \frac{\frac{\pi}{2} - 0}{5 \, \text{s}} = 0.314 \, \text{rad/s}^2\] Being aware of angular acceleration is crucial for predicting motion in rotating systems and ensuring smooth operation across various applications.