10.76.
Model:
Model the sled as a particle. Because there is no friction, the sum of the kinetic and
gravitational potential energy is conserved during motion.
Visualize:
Place the origin of the coordinate system at the center of the hemisphere. Then
0
yR
=
and, from geometry,
1
y
=
cos .
R
φ
Solve:
The energy conservation equation
11 0 0
K
UKU
+
=+
is
22
2
11
00
1
1
1
cos
2
(1 cos )
2
mv
mgy
mv
mgy
mv
mgR
mgR
v
gR
φφ
+=
+⇒
+
=⇒
=
−
(b)
If the sled is on the hill, it is moving in a circle and the
r
component of
net
F
G
has to point to the center with
magnitude
2
net
/.
Fm
v
R
=
Eventually the speed gets so large that there is not enough force to keep it in a circular
trajectory, and that is the point where it flies off the hill. Consider the sled at angle
.
Establish an
r
axis
pointing toward the center of the circle, as we usually do for circular motion problems. Newton’s second law
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This note was uploaded on 12/10/2011 for the course PHYS 1211 taught by Professor Geller during the Fall '09 term at UGA.
 Fall '09
 geller
 Energy, Friction, Potential Energy

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