# A washing machine has a fast spin cycle of 542 rev/min and a slow spin cycle of 328 rev/min. The diameter of the washing machine drum is 0.43 m. What is the ratio of the centripetal accelerations for the spin cycles?

Jun 15, 2017

$\text{Fast : slow}$ $=$ $2.73$ $: 1$

#### Explanation:

The equation for the centripetal acceleration ${a}_{\text{rad}}$ of an object in uniform circular motion is given by

${a}_{\text{rad}} = \frac{4 {\pi}^{2} R}{{T}^{2}}$

where

• $R$ is the radius of the circle of motion, which in this case is $\frac{0.43 \text{m}}{2} = 0.215$ $\text{m}$

• $T$ is the time for one revolution, in $\text{s}$. To find this, we take the given untis of "revolutions per minute", convert it to "revolutions per second", and find the reciprocal of that (to get "seconds per revolution"):

T_1 = ((542"rev")/(1cancel("min")))((1cancel("min"))/(60"s")) = 9.03"rev"/"s" = overbrace(0.111"s")^("reciprocal of"color(white)(x) 9.03)

T_2= ((328"rev")/(1cancel("min")))((1cancel("min"))/(60"s")) = 5.47"rev"/"s" = overbrace(0.183"s")^("reciprocal of"color(white)(x) 5.47)

Plugging in the known values, we have, for each acceleration:

a_"rad-1" = (4pi^2(0.215"m"))/((0.111"s")^2) = color(red)(693 color(red)("m/s"^2

a_"rad-2" = (4pi^2(0.215"m"))/((0.183"s")^2) = color(blue)(254 color(blue)("m/s"^2

The ratio of the centripetal acceleration of the fast-speed setting to the low-speed setting is thus

color(red)(693 color(red)("m/s"^2 :color(blue)(254 color(blue)("m/s"^2) = color(darkorange)(2.73 color(darkorange)( :1

We can determine (from the equation) that the ratio is equal to the inverse of the square of the first time ${T}_{1}$ to the inverse of the square of the second time ${T}_{2}$:

${a}_{\text{rad"-1): a_("rad} - 2} = \frac{1}{{\left({T}_{1}\right)}^{2}} : \frac{1}{{\left({T}_{2}\right)}^{2}}$

1/((T_1)^2): 1/((T_2)^2) = 1/(0.111"s"^2): 1/(0.183"s"^2) = color(darkorange)(2.73 $\textcolor{\mathrm{da} r k \mathmr{and} a n \ge}{: 1}$