Question

during the spin cycle of your clothes washer, the tub rotates at a steady angular velocity of 38.9 rad/s. determine the angular displacement δθ of the tub during a spin of 67.3 s in units of radians and revolutions. δθ = \boxed{ } rad \quad δθ = \boxed{ } rev

Explanation:

Step1: Recall the formula for angular displacement

The formula for angular displacement \(\Delta\theta\) when angular velocity \(\omega\) is constant is \(\Delta\theta=\omega\times t\), where \(\omega\) is the angular velocity and \(t\) is the time.
Given \(\omega = 38.9\space rad/s\) and \(t = 67.3\space s\).

Step2: Calculate angular displacement in radians

Substitute the values into the formula:
\(\Delta\theta=\omega\times t=38.9\times67.3\)
\(38.9\times67.3 = 38.9\times(60 + 7+ 0.3)=38.9\times60+38.9\times7 + 38.9\times0.3=2334+272.3+11.67 = 2617.97\space rad\) (approx)

Step3: Convert radians to revolutions

We know that \(1\space revolution = 2\pi\space radians\), so the number of revolutions \(n=\frac{\Delta\theta}{2\pi}\)
Substitute \(\Delta\theta = 2617.97\space rad\)
\(n=\frac{2617.97}{2\pi}\approx\frac{2617.97}{6.2832}\approx416.66\space rev\)

Answer:

For radians: \(\Delta\theta\approx\boldsymbol{2618}\space rad\) (rounded to a reasonable number, more precisely \(2617.97\))
For revolutions: \(\Delta\theta\approx\boldsymbol{417}\space rev\) (rounded, more precisely \(416.66\))

(If we use more precise calculation for the first step: \(38.9\times67.3 = 38.9\times67.3 = 2617.97\) rad. Then for revolutions: \(\frac{2617.97}{2\times3.1416}\approx\frac{2617.97}{6.2832}\approx416.66\) rev)