A uniform soild right circular cylinder of weight 2W and radius r is cut into two equal parts by a plane through its axis to form two semicircular cylinders. One of these semicircular cylinders is placed with its curved surface on a plane inclined at an angle @ to the horizontal
(@<= arc sin(4/3pi) )
such that the generators of hte cylinder are horizontal. Suppose that the base of the semicircular cylinder is inclined at an angle x (x<=90 degree) to the horizontal and that the contact between the curved surface and the inclined plane is sufficiently rough to prevent any slipping.
(a) Prove that, in a position of equilibrium
x = arc sin(3pi sin@ / 4)
(b) A particle of weight 4W/9pi is then placed on the base at the lowest tip B. If @ = 10 degree , prove that a new position of equilibrium, without the particle slipping, is possible, if the coefficient between the particle and the base is 1.0 .
Applied Maths - Mechanics 16
2009-07-31 2:37 am
回答 (2)
2009-08-02 2:21 am
✔ 最佳答案
Do not understand the question, do you have a diagram?2009-08-01 18:21:20 補充：
Consider the triangle ABC, where B is the centre of gravity of the semi-cylinder, B is the contact point of the semi-cylinder to the slope and C the centre of the circle.
http://img150.imageshack.us/img150/5188/semicyl1.jpg
By sine rule,
CA / sinABC = CB / sinCAB
r / sin(180 - x) = 4r/3π / sinθ
sinθ = 4/3πsinx
sinx = 3πsinθ/4
x = sin-13πsinθ/4
When the particle is added, the semi-cylinder will turn a bit anti-clockwise.
http://img76.imageshack.us/img76/9985/semicyl2.jpg
In the diagram, the dimensions are:
h = rcosx
k = rsinθ
k + p = (4r/3π)sinx
p = (4r/3π)sinx - rsinθ
Take moment about the new contact point A,
(4W/9π)(h + k) = Wp
(4W/9π)(rcosx + rsinθ) = W[(4r/3π)sinx – rsinθ]
(4/9π)(cosx + sinθ) = [(4/3π)sinx – sinθ]
0.1415cosx + 0.0246 = 0.4244sinx – 0.1736
0.1415cosx – 0.4244sinx = – 0.1736 – 0.0246
let tany = 0.4244/0.1415
so y = 71.56
0.4474(cosxcosy – sinxsiny) = - 0.1982
cos(x+y) = -0.443
x + y = 116.30
x = 44.74
Assume equilibrium is possible
Z = weight of particle = 4W/9π
N = Zcosx
F = Zsinx
F/N = tanx must be smaller than the coefficient of friction
Coefficient of friction > tanx = 1.0
2009-08-01 7:09 pm
For image , please refer to http://img190.imageshack.us/img190/7418/cylinderg.jpg
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