A sphere of mass m is placed on a rough block of mass 2m which in turn is placed on a rough inclined plane. Friction coefficient between block and sphere isand that between block and the inclined plane is
. Find the maximum value of
to ensure pure rolling at block and sphere surface.
This question, as it is currently stated, makes no sense!
No statement is made about whether the block is at the point of slipping or is slipping and so to maximise the ratio one can assume thatis sufficiently large that the block remains where it is. In this case there is only a fixed condition on the minimum of
and as we can set
Probably we are supposed to assume this:
Find the maximum value ofto ensure pure rolling at block and sphere surface, and sliding at block and plane surface.
If so, here is the solution (just tentative, you are very welcome to point out if there is error):
For the spherical ball,
=frictional force
=contact force
=moment of inertia
=angular acceleration
=translational acceleration along the surface
(1)
(2)
(3)
For pure rolling:
(4)
(5)
From (1)~(5),
(6)
(7)
(8)
For the block,
=frictional force along the block-plane interface
=contact force along the block-plane interface
=mass of block,
(9)
(10)
(11)
From (6)(7) and (9)~(11),
(12)
From (8) and (12),
(13)
Let, so
(14)
Notefor solid sphere,
for spherical shell,
for solid cylinder,
for cylindrical shell,. So finally
for solid sphere.
for spherical shell.
for solid cylinder.
for cylindrical shell.
I am also wondering if there exists another possibility, i.e. the block goes up along the plane (rather than go down). If so, equation (9) needs to be changed. Interested persons might investigate that possibility.
Luobo
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Thats why I posted it. Are we supposed to assume something? 