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Chapter 9: Problem 18

Express \(40.0 \mathrm{deg} / \mathrm{s}\) in \((a) \mathrm{rev} / \mathrm{s},(b) \mathrm{rev} / \mathrm{min}\), and \((c) \mathrm{rad} / \mathrm{s}\).

Short Answer

Expert verified
(a) 0.111 rev/s, (b) 6.67 rev/min, (c) 0.698 rad/s.

Step by step solution

01

Convert Degrees per Second to Revolutions per Second

To convert from degrees per second to revolutions per second, use the fact that one revolution is equal to 360 degrees. Thus:\[ \frac{40.0 \, \text{deg}}{\text{s}} \times \frac{1 \, \text{rev}}{360 \, \text{deg}} = \frac{40.0}{360} \, \text{rev/s} = \frac{1}{9} \, \text{rev/s} \approx 0.111 \, \text{rev/s} \]
02

Convert Revolutions per Second to Revolutions per Minute

Since there are 60 seconds in a minute, multiply the result from the previous step by 60 to convert to revolutions per minute:\[ 0.111 \, \text{rev/s} \times 60 \, \text{s/min} = 6.67 \, \text{rev/min} \]
03

Convert Degrees per Second to Radians per Second

Since there are \(2\pi\) radians in a full circle (360 degrees), convert degrees to radians as follows:\[ \frac{40.0 \, \text{deg}}{\text{s}} \times \frac{2\pi \, \text{rad}}{360 \, \text{deg}} = \frac{40.0 \times \pi}{180} \, \text{rad/s} = \frac{2\pi}{9} \, \text{rad/s} \approx 0.698 \, \text{rad/s} \]

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Key Concepts

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

Degrees to Radians Conversion
Degrees and radians are both units for measuring angles, but radians are based on the radius of a circle. To convert an angle from degrees to radians, you use the fact that 360 degrees equals \(2\pi\) radians. Here's how the conversion works:
  • Understand that \(360\) degrees equals \(2\pi\) radians; thus, \(180\) degrees equals \(\pi\) radians.
  • The formula for converting degrees to radians is: \(\theta \text{ (in radians)} = \theta \text{ (in degrees)} \times \frac{\pi}{180}\).
  • This formula arises from knowing that \( \frac{180}{\pi} \) in degrees corresponds to \(1\) radian.
Thus, when you convert 40 degrees per second to radians per second, you multiply \(40\) by \( \frac{\pi}{180} \), resulting in approximately \(0.698\) radians per second. Converting between these two units frequently is crucial in physics for understanding rotational motion.
Revolutions per Second
Revolutions per second (rev/s) is a measure of how many full turns occur each second. It's a basic unit for expressing angular velocity, making it very useful in problems involving circular motion.
  • A complete revolution equals \(360\) degrees or \(2\pi\) radians.
  • To convert from revolutions to other units, always remember this relationship.
  • The conversion between degrees per second and revolutions per second uses the relationship \(1 \text{ rev} = 360 \text{ deg}\).
For example, converting \(40\) degrees per second to revolutions per second involves taking the degree amount \(40\), dividing by \(360\), resulting in \( \frac{1}{9} \) rev/s or approximately \(0.111\) rev/s. This conversion helps understand how quickly an object is rotating.
Radians per Second
The unit of radians per second (rad/s) measures angular velocity in radians over a single second. It's especially important when dealing with rotational systems in physics, where radians are a natural choice due to their relationship with the circumference and radius of a circle.
  • One revolution is \(2\pi\) radians, equivalent to \(360\) degrees.
  • To convert from degrees per second to radians per second, multiply the degree value by \(\frac{\pi}{180}\).
  • Utilizing radians makes calculations in rotational dynamics simpler and more intuitive.
For instance, when you transform \(40\) degrees per second to radians per second, you apply the conversion, yielding approximately \(0.698\) rad/s. Radians per second give a direct measure of angular displacement over time.
Angular Velocity Conversion
Angular velocity is the rate of change of angle per unit of time. It is crucial to express this velocity in different units depending on the context of a problem. Conversions are commonplace in physics to aid in understanding and applying rotational motion concepts.
  • Typically expressed in degrees per second, revolutions per second, or radians per second.
  • Conversions allow us to interchangeably use these units to solve problems and better grasp angular relations.
  • For example, the conversion from degrees per second to radians per second or revolutions per minute enhances the understanding of the motion of rotating objects.
Transforming between these units is fundamental for connecting different rotational dynamics problems. For example, given \(40.0\) degrees per second, you can find angular velocity in revolution-based terms by sequentially converting to rev/s and rev/min. This understanding facilitates problem-solving across various physics contexts.

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Most popular questions from this chapter

( \(a\) ) Compute the radial acceleration of a point at the equator of the Earth. (b) Repeat for the North Pole of the Earth. Take the radius of the Earth to be \(6.37 \times 10^{6} \mathrm{~m}\).

Through how many radians does a point fixed on the Earth's surface (anywhere off the poles) move in \(6.00 \mathrm{~h}\) as a result of the Earth's rotation? What is the linear speed of a point on the equator? Take the radius of the Earth to be \(6370 \mathrm{~km}\).

The bob of a pendulum \(90 \mathrm{~cm}\) long swings through a \(15-\mathrm{cm}\) arc, as shown in Fig. \(9-3\). Find the angle \(\theta\), in radians and in degrees, through which it swings. Fig. \(9-3\) Recall that \(l=r \theta\) applies only to angles in radian measure. Therefore, in radians $$ \theta=\frac{l}{r}=\frac{0.15 \mathrm{~m}}{0.90 \mathrm{~m}}=0.167 \mathrm{rad}=0.17 \mathrm{nd} $$ Then in degrees \(\quad \theta=\left(0.167\right.\) nd) \(\left(\frac{360 \mathrm{deg}}{2 \pi \mathrm{ral}}\right)=9.6^{\circ}\)

A satellite orbits the Earth at a height of \(200 \mathrm{~km}\) in a circle of radius \(6570 \mathrm{~km}\). Find the linear speed of the satellite and the time taken to complete one revolution. Assume the Earth's mass is \(6.0 \times 10^{24} \mathrm{~kg} .\) [Hint: The gravitational force provides the centripetal force.]

A car wheel \(30 \mathrm{~cm}\) in radius is tuming at a rate of \(8.0 \mathrm{rev} / \mathrm{s}\) when the car begins to slow uniformly to rest in \(\mathrm{A}\) time of \(14 \mathrm{~s}\). Find the number of revolutions made by the wheel and the distance the car goes in the \(14 \mathrm{~s}\).
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