Question 2177: Dear sir,
It is not a hollow cylinder .It is a verticallly positioned rotating cylinder(i.e. cylinder in sleeping position ,or in other words, the cylinder height is parallel to the earth suface).It is filled with balls (of different dimensions) to a particular height inside the mill. Company people will not give the amount of balls in the mill.they will provide only the height upto which the balls are present inside the mill.they will not give other data except this.
So my data are follows,
Mill dia = 2.21m
Mill Length = 3.21m
Height upto which the balls are present inside the mill = 1.14m
SO the free space inside the mill is (2.21(dia) - 1.14) = 1.07 m
SO I have considered this part as a segment area. Hope now you can understand the concept.
Please reply the various formulae for Segment Area soon
Thankyou,
Prem
Answer by longjonsilver(2297)   (Show Source):
You can put this solution on YOUR website! ok, got it :-)
Let centre of circle be O, Let height of balls be a horizontal line AB, which is 1.14m above the bottom of the cyclinder (and is above the centre of the circle) since cylinder radius = 1.105m.
I shall consider the free space above the balls...this is the segment made by the line AB, as part of the sector AOB. Given that the angle AOB is a (in radians!!), then we know the following formulae:
area of sector AOB = (1/2)r^2a
area of triangle AOB = (1/2)r^2sina
so area of segment made by AB is area of sector - area of triangle
segment = (1/2)r^2a - (1/2)r^2*sina
segment = (1/2)r^2(a-sina)
We need the angle a!
Draw a vertical line up from O to the mid-point of AB, call this point C. Now look at the right angled triangle AOC....
OA = radius = 1.105m
OC = 1.14-1.105m = 0.035m
angle AOC = b
so cosb=OC/OA
therefore b = 88.18489degrees
so angle a = 2b = 176.3697 degrees.
Now to convert this to radians....
360 degree = 2pi radians
so 1 degree is (2pi)/360 radians
hence 176.36 degrees is 3.07823 radians (to however many dp you want)
so,
segment above balls = (1/2)(1.105)^2(3.07823-sin(3.07823))
= 1.8406m^2.
As cylinder area = pi*r^2
total Area = 3.83596m^2
then area of balls = 3.83596-1.8406 = 1.99546m^2
Hope this helps...Please check my working though, as i was rushing (i am at work), but the approach is correct.
jon.
|
|
|
