Physics 2211
Quiz #4
Solutions
Spring 2008
Unless otherwise directed, all springs, cords, and pulleys are ideal, and drag should be neglected.
I
. (16 points) Two blocks of mass
m
1
and
m
2
are tied together and pulled so that it travels up a frictionless
incline that makes an angle
θ
with the horizontal, as illustrated.
The force pulling them is a tension of
magnitude
T
1
, and their resulting acceleration is not zero.
What is the tension magnitude
T
2
in rope 2,
which ties the blocks together, in terms of any or all of
m
1
,
m
2
,
θ
,
T
1
, and physical or mathematical
constants? (
On Earth.
)
.
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There is more than one approach to solving this Newton’s Second Law problem.
The easiest is probably
to look first at the two blocks
m
1
and
m
2
along with rope 2 as a single object being pulled up the incline
by rope 1. A FreeBody Diagram has been sketched for this object. A coordinate system has been chosen
so the acceleration on one axis is zero, and the weight force has been resolved into components. Applying
Newton’s Second Law to the
x
direction:
X
F
x
=
T
1

w
tot
x
=
m
tot
a
x
⇒
T
1

(
m
1
+
m
2
)
g
sin
θ
= (
m
1
+
m
2
)
a
x
a
x
=
T
1

(
m
1
+
m
2
)
g
sin
θ
m
1
+
m
2
Looking next at just the block with mass
m
2
, which is pulled up the ramp by rope 2, a FreeBody Diagram
has been sketched for this object. The same coordinate system is used, so the acceleration of this block will
be represented by the same symbol as the acceleration of the two blocks together. The weight force has been
resolved into components. Again, applying Newton’s Second Law to the
x
direction:
X
F
x
=
T
2

w
2
x
=
m
tot
a
x
⇒
T
2

m
2
g
sin
θ
=
m
2
a
x
Substituting the expression for the acceleration and solving for
T
2
:
T
2

m
2
g
sin
θ
=
m
2
T
1

(
m
1
+
m
2
)
g
sin
θ
m
1
+
m
2
¶
⇒
T
2
=
m
2
T
1

(
m
1
+
m
2
)
g
sin
θ
m
1
+
m
2
¶
+
m
2
g
sin
θ
That is sufficient to answer the question. A bit more algebra, though, yields an interesting result:
T
2
=
m
2
T
1

(
m
1
+
m
2
)
g
sin
θ
m
1
+
m
2
¶
+
m
1
+
m
2
m
1
+
m
2
¶
m
2
g
sin
θ
=
m
2
T
1

(
m
1
+
m
2
)
g
sin
θ
+ (
m
1
+
m
2
)
g
sin
θ
m
1
+
m
2
¶
=
m
2
m
1
+
m
2
¶
T
1
Quiz #4 Solutions Page 1 of 5
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II
. (16 points) A physics book of mass
M
is connected by a string to a coffee cup of mass
m
. The book is given
a push up a slope that makes an angle
θ
with the horizontal, and released at initial speed
v
i
, as illustrated.
The coefficient of static friction between the book and the slope is
μ
s
, while the coefficient of kinetic friction
is
μ
k
. How much time is required for the book to come to a stop after it is released, in terms of any or all
of
M
,
m
,
θ
,
μ
s
,
μ
k
, and physical or mathematical constants? (
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 Spring '08
 UZER
 Force, Friction, Mass, Solutions Page, Fneg

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