MATLAB Problem: HalfPipe
In this problem you will analyze the motion of a
block on a halfcircle depicted in the figure. The
dashed block shows the initial position. It is
released from rest. The coefficient of kinetic
friction is
, the mass is
m
, the circle radius is
r
,
the gravitational constant is
g
.
Let
be the angle of rotation from the
Cartesian coordinate basis
to the
polar coordinate basis
.
1)
What is the relation between
and
?
2)
Derive the linear transformation from
to
in terms of
.
3)
Show the linear transformation derived in 2) is a rotation matrix.
4)
Derive expressions for the forces acting on the block in polar coordinates, in terms of
m
,
g
,
r
,
,
,
,
N
(the normal force in the direction of
).
5)
Derive an expression for the acceleration of the block in polar coordinates using
(instead of
).
6)
From the balance of linear momentum (combining 4) and 5)), derive an expression for
in
terms of
g
,
r
,
,
,
.
7)
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 Fall '08
 Oreilly
 Force, Mass, Total Energy

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