/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 87 The angular velocity of a wheel ... [FREE SOLUTION] | 91影视

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Chapter 3: Problem 87

The angular velocity of a wheel increases from 1200 rpm to 4500 rpm in \(10 \mathrm{~s}\). The number of revolutions made during this time is (A) 950 (B) 475 (C) \(237.5\) (D) \(118.75\)

Short Answer

Expert verified
The short answer is: (A) 950.

Step by step solution

01

Convert given angular velocities to radians per second.

We are given the angular velocities in revolutions per minute (rpm), so we need to convert them to radians per second for better calculations. To convert rpm to radians per second, we use the formula: \[\omega_{rad} = \omega_{rpm} \times \frac{2\pi}{60}\] where \(\omega_{rad}\) represents angular velocity in radians per second and \(\omega_{rpm}\) represents angular velocity in revolutions per minute. For initial angular velocity: \[\omega_1 = 1200 \times \frac{2\pi}{60} = 40\pi\hspace{1cm}(1)\] For final angular velocity: \[\omega_2 = 4500 \times \frac{2\pi}{60} = 150\pi\hspace{1cm}(2)\]
02

Calculate the average angular velocity.

Now we need to find the average angular velocity during the acceleration. Since the acceleration is constant, we can take an average of the initial and final angular velocities. Average angular velocity \(\omega_{avg}\) can be calculated as follows: \[\omega_{avg} = \frac{\omega_1 + \omega_2}{2}\] Using the values from equations (1) and (2): \[\omega_{avg} = \frac{40\pi + 150\pi}{2} = \frac{190\pi}{2} = 95\pi\]
03

Calculate the number of revolutions in the given time.

We are given that the duration of acceleration is 10 seconds. To find the number of revolutions during this time, we need to multiply the average angular velocity by the time and divide by \(2\pi\). Number of revolutions: \[N = \frac{\omega_{avg} \times t}{2\pi}\] Using the value of \(\omega_{avg}\) and given time: \[N = \frac{95\pi \times 10}{2\pi} = 950\] So, the number of revolutions made by the wheel during this time is: (A) 950

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