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Chapter 10: Problem 15

Express each of the following in radians per second: (a) 720 rpm: (b) \(50^{\circ} \mathrm{M} ;\) (c) 1000 rev/s; (d) 1 rev/year (Earth's angular speed in its orbit).

Short Answer

Expert verified
(a) 75.39822369 rad/s; (b) 0.01460776942 rad/s; (c) 6283.18530718 rad/s; (d) \(1.99228302 \times 10^{-7} \, \text{rad/s}\)

Step by step solution

01

Convert rev/min to rad/s - 720 rpm

For part (a), we need to convert 720 rev/min into radians per second. First, convert minute to second using the conversion 1 min = 60s. Then, convert revolutions to radians using 1 rev = \(2\pi\) rad. The total conversion is: \(720 \, \text{rev/min} = 720 (\frac{2\pi \, \text{rad}}{1 \, \text{rev}})(\frac{1 \, \text{min}}{60 \, \text{s}}) = 75.39822369 \, \text{rad/s}\)
02

Convert degrees/min to rad/s - \(50^{\circ}\) /min

For part (b), we convert \(50^{\circ}/\text{min}\) into radians per second. Convert the degree to radian using \(\pi\) rad = \(180^{\circ}\) and minutes to seconds with 1 min = 60s. Hence, the conversion is: \(50^{\circ}/\text{min} = 50 (\frac{\pi \, \text{rad}}{180^{\circ}})(\frac{1 \, \text{min}}{60 \, \text{s}}) = 0.01460776942 \, \text{rad/s}\)
03

Convert rev/s to rad/s - 1000 rev/s

Part (c) involves converting from revolutions per second to radians per second. The conversion is quite straightforward: \(1000 \, \text{rev/s} = 1000 (\frac{2\pi \, \text{rad}}{1 \, \text{rev}}) = 6283.18530718 \, \text{rad/s}\)
04

Convert rev/year to rad/s - 1 rev/year

For the last unit conversion in part (d), we have 1 revolution per year. Use the fact that 1 year has 31.536 million seconds and 1 rev = \(2\pi\) rad. The conversion is: \(1 \, \text{rev/year} = 1 (\frac{2\pi \, \text{rad}}{1 \, \text{rev}})(\frac{1 \, \text{yr}}{31.536 \times 10^6 \, \text{s}}) = 1.99228302 \times 10^{-7} \, \text{rad/s}\)

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