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?

The equation for the centripetal acceleration #a_"rad"# of an object in uniform circular motion is given by

#a_"rad" = (4pi^2R)/(T^2)#

where

• #R# is the radius of the circle of motion, which in this case is #(0.43"m")/2 = 0.215# #"m"#

• #T# is the time for one revolution, in #"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_("rad"-1): a_("rad"-2) = 1/((T_1)^2): 1/((T_2)^2)#

#1/((T_1)^2): 1/((T_2)^2) = 1/(0.111"s"^2): 1/(0.183"s"^2) = color(darkorange)(2.73# #color(darkorange)( :1)#