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Chapter 6: Problem 51

A wheel is rotating at 900 rpm about its axis. When power is cut off it comes to rest in 1 minute. The angular retardation is \(\frac{\pi}{n} \mathrm{rad} / \mathrm{s}^{2}\), then the value of \(n\) is.

Short Answer

Expert verified
The initial angular velocity is \(30\pi \, \text{rad/s}\). Using the equation \(\omega_f = \omega_i - \alpha t\) and given angular retardation \(\frac{\pi}{n} \, \text{rad/s}^2\), solving for n gives us the value of n as 2.

Step by step solution

01

Convert rpm to radians per second

First, we need to convert the given initial angular velocity (900 rpm) to radians per second. Recall that 1 revolution is equal to \(2\pi\) radians and 1 minute is equal to 60 seconds. Therefore, we can convert 900 rpm to radians per second using the following equation: \[\omega_i = \frac{900 \, \text{rev}}{\text{min}} \times \frac{2\pi \, \text{rad}}{\text{rev}} \times \frac{1 \, \text{min}}{60\, \text{s}}\]
02

Calculate initial angular velocity in radians per second

Now, let's calculate the initial angular velocity in radians per second: \[\begin{aligned} \omega_i &= \frac{900 \times 2\pi}{60} \\ &= \frac{1800\pi}{60} \\ &= 30\pi \, \text{rad/s} \end{aligned}\] The initial angular velocity of the wheel is \(30\pi \, \text{rad/s}\).
03

Use the angular retardation equation

Next, we will use the equation for angular retardation: \[\omega_f = \omega_i - \alpha t\] where \(\omega_f\) is the final angular velocity, \(\omega_i\) is the initial angular velocity, \(\alpha\) is the angular retardation, and \(t\) is the time. In this case, the wheel comes to rest (\(\omega_f = 0\)), the initial angular velocity is \(30\pi \, \text{rad/s}\), and the time is 1 minute (60 seconds). We are given that the angular retardation is \(\frac{\pi}{n} \, \text{rad/s}^2\), so we can write the equation as: \[0 = 30\pi - \frac{\pi}{n} \cdot 60\]
04

Solve for n

Now, let's solve for n: \[\begin{aligned} 0 &= 30\pi - \frac{60\pi}{n} \\ \frac{60\pi}{n} &= 30\pi \\ n &= \frac{60}{30} \\ n &= 2 \end{aligned}\] The value of n is 2.

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