Physics 2211
Quiz #5
Solutions
Spring 2008
Unless otherwise directed, all springs, cords, and pulleys are ideal, and drag should be neglected.
I
. (16 points) An amusement park sled and rider, with total mass
m
, is to be sent around a looptheloop of
radius
R
. The sled and rider will accomplish this by compressing a large spring with Hooke’s Law constant
k
, and being released from rest. The loop is frictionless, but the level ground has a coeﬃcient of kinetic
friction
μ
k
with the sled. After the spring has been compressed a distance
s
, the sled is a distance
d
from the
loop. At what maximum distance,
d
, from the loop can the sled be released if the loop is to be completed?
Express your answer in terms of any or all of
m
,
R
,
k
,
μ
k
, and physical or mathematical constants? (
On
Earth.
)
.
.
.
.
.
.
.
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.
.
Announced during quiz: Your answer may depend on
s
.
Use the WorkEnergy Theorem. Consider the Earthspringcar sys
tem. Elastic (spring) and gravitational potential energies change
within the system. Friction is a nonconservative force doing work
within the system. There are no external forces doing work on the
system.
W
ext
+
W
nc
= Δ
K
+ Δ
U
= (
K
f

K
i
) + (
U
s
f

U
s
i
) + (
U
g
f

U
g
i
)
0 +
Z
~
f
·
d
~
d
=
(
1
2
mv
2
f

1
2
mv
2
i
)
+
‡
1
2
k
(Δ
s
f
)
2

1
2
k
(Δ
s
i
)
2
·
+ (
mgh
f

mgh
i
)
The frictional force,
~
f
, has constant magnitude
μ
k
n
, and is opposite the displacement. Let the zero of height
be at ground level (the initial position). The positive direction must be up and the zero of spring compression
be at the relaxed (ﬁnal) position to use the expressions above. The initial compression of the spring is
s
,
and the initial speed of the car is zero. The height of the car at the top of the loop is 2
R
.
μ
k
nd
cos180
◦
=
(
1
2
mv
2
f

0
)
+
(
0

1
2
ks
2
)
+ (
mg
2
R

0)
The normal force while the car is on the ground and friction is acting can be found from Newton’s Second
Law. The car has no vertical (
y
) acceleration at that point.
X
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 Spring '08
 UZER
 Energy, Force, Kinetic Energy, Mass, Potential Energy, Total mechanical energy, Solutions Page

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