yindeemark (rry82) – Homework 12 – chelikowsky – (59005)
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001
10.0 points
Assume:
A bullet of mass
m
and cube of
mass
M
undergo an inelastic collision, where
m
≪
M
.
Note:
The moment of inertia of this cube
(with edges of length 2
a
and mass
M
) about
an axis along one of its edges is
8
M a
2
3
.
A solid cube is resting on a horizontal sur
face. The cube is constrained to rotate about
an axis at its bottom right edge (due to a
small obstacle on the table).
A bullet with
speed
v
min
is shot at the lefthand face at a
height of
4
3
a
.
The bullet gets embedded in
the cube.
2
a
M
mvectorv
min
4
3
a
ω
M g
Find the minimum value of
v
min
required
to tip the cube so that it falls its righthand
face.
1.
v
min
=
m
M
radicalbigg
5
g a
parenleftBig
√
2
−
1
parenrightBig
2.
v
min
=
M
m
radicalbigg
3
g a
parenleftBig
√
2
−
1
parenrightBig
correct
3.
v
min
=
m
M
radicalbigg
2
g a
parenleftBig
√
2
−
1
parenrightBig
4.
v
min
=
m
M
radicalbigg
3
g a
parenleftBig
√
5
−
1
parenrightBig
5.
v
min
=
m
M
radicalbigg
3
g a
parenleftBig
√
3
−
1
parenrightBig
Explanation:
Basic Concepts:
summationdisplay
vector
L
=
const
Δ
U
+ Δ
K
= 0
For the cube to tip over the center of mass
(CM) must rise so that it is over the axis of
rotation
AB
.
To do this the CM must be
raised a distance of
a
parenleftBig
√
2
−
1
parenrightBig
.
From conservation of energy
M g a
parenleftBig
√
2
−
1
parenrightBig
=
1
2
I
cube
ω
2
.
(1)
From conservation of angular momentum
4
a
3
m v
=
parenleftbigg
8
M a
2
3
parenrightbigg
ω
ω
=
parenleftBig
m v
2
M a
parenrightBig
.
(2)
Thus, substituting Eq. 2 into 1, we have
1
2
parenleftbigg
8
M a
2
3
parenrightbigg parenleftbigg
m
2
v
2
4
M
2
a
2
parenrightbigg
=
M g a
parenleftBig
√
2
−
1
parenrightBig
Solving for
v
yields
v
min
=
M
m
radicalbigg
3
g a
parenleftBig
√
2
−
1
parenrightBig
.
002
10.0 points
If you walk along the top of a fence, why does
holding your arms out help you to balance?
1.
Your momentum is decreased.
2.
Your rotational inertia is decreased.
3.
Your rotational inertia is increased.
cor
rect
4.
Your momentum is increased.
Explanation:
Your rotational inertia increases when your
arms are outstretched, which increases the
resistance to a change in your rotational state.
003
10.0 points
The angle of the inclination is 34
.
9
◦
, the outer
part of the large pulley has a radius of 3
r
and
the inner part of the large pulley has a radius
of
r
.
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yindeemark (rry82) – Homework 12 – chelikowsky – (59005)
2
3
r
r
/2
r
m
T
θ
Find the mass
m
needed to balance the
820 kg truck on the incline. The acceleration
of gravity is 9
.
8 m
/
s
2
.
Assume all pulleys are
frictionless and massless and the cords are
massless.
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 Fall '08
 Turner
 Angular Momentum, Mass, Simple Harmonic Motion, Work, Correct Answer

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