A 5.0cm diameter steel shaft has 0.10mm clearance all around its bushing at 20 degree Celsius.
If the bushing temperature remains constant, at what temperature will the shaft begin to bind? Steel has linear expansion coefficient of 11x10^6/degree Celsius.
A) 353 degree Celsius
B) 333 degree Celsius
C) 53 degree Celsius
D) 680 degree Celsius
The measrement of the shaft diameter at 20 C is 5.cm. Since it has clearance of 0.1mm around, the diameter has to expand by 0.1mm*2 = 0.2mm =0.02cm.
Since the coefficient of linear expansion of steal is 11*10^-6 /C, the expansion by x degree raise in temperature raise is = 5cm*(11*10^-6)*x and this should be equal to 0.02cm.Therefore,
(5cm)(11*10^-6)x = 0.02cm. Solving for x,
x=0.02/(5cm)(11*10^-6) = 363.63 to be raised from 20 degree.
The shaft will begin to bind when due to increasing temperature the diameter of shaft will increase to cover completely the clearance between the shaft and the bushing.
Given shaft diameter = D = 5 cm = 50 mm
Clearance between shaft and bushing = 0.1 mm all around
Initial temperature of shaft = t = 20 degree C
Linear coefficient of expansion = E = 11*18^-6 /degree C
As the clearance between shaft and bushing is all around, the increase in diameter of shaft (d) when the shaft just fill the clearance completely is twice the clearance. Thus:
d = 2*Clearance = 2*0.1 = 0.2 mm
The temperature rise (T) when the diameter D will expand by d is given by formula:
T = (d/D)/E = (0.2/50)/(11*10^-6) = 363.63 C
Temperature of shaft when it begins to bind is initial temperature (t) plus increase in temperature (T).
Therefore temperature of shaft = t +T = 20 + 363.63 = 383.63
= 383 C (rounding off to nearest unit number)
The shaft will begin to bind when its temperature is 383 C.
None of the options given in the question corresponds to this.
Perhaps, by mistake, temperature given in option B) is given as 333 instead of 383.