An elevator cable winds on a drum of radius \(90.0 \mathrm{cm}\) that is connected to a motor. (a) If the elevator is moving down at $0.50 \mathrm{m} / \mathrm{s},$ what is the angular speed of the drum? (b) If the elevator moves down \(6.0 \mathrm{m},\) how many revolutions has the drum made?

First, we are given the radius of the drum (\(r = 90\ \mathrm{cm} = 0.9\ \mathrm{m}\)) and the linear speed of the elevator (\(v = 0.50\ \mathrm{m/s}\)). We can use the relationship \(v = ωr\) to find the angular speed (\(ω\)) of the drum. Rearrange the formula to solve for \(ω\): \(ω = \frac{v}{r}\) Now, substitute the given values into the formula: \(ω = \frac{0.5\ \mathrm{m/s}}{0.9\ \mathrm{m}}\) Calculate: \(ω = 0.56\ \mathrm{rad/s}\) So, the angular speed of the drum is \(0.56 \mathrm{rad/s}\).

Next, we are asked to find how many revolutions the drum has made when the elevator moves \(6.0\ \mathrm{m}\). First, we need to find the length of the cable that has been wound on the drum when the elevator moves down \(6.0\ \mathrm{m}\). The length of the cable wound can be found by multiplying the distance the elevator moves down (\(6.0\ \mathrm{m}\)) with the circumference of the drum (\(2πr\)). Then, divide the result by the circumference of the drum to find the number of revolutions. Length of cable wound on the drum: \(Length = 2πr * \mathrm{revolutions}\) Revolutions can be found by: \(\mathrm{revolutions} = \frac{Length}{2πr}\) Now, substitute the given values into the formula: \(\mathrm{revolutions} = \frac{6.0\ \mathrm{m}}{2π(0.9\ \mathrm{m})}\) Calculate: \(\mathrm{revolutions} = 1.06\) Therefore, the drum has made approximately \(1.06\) revolutions when the elevator moves down \(6.0\ \mathrm{m}\).