alexander (jra2623) – homework 21 – Turner – (92510)
1
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001
10.0 points
A uniform rod of mass 2
.
7 kg is 7 m long. The
rod is pivoted about a horizontal, frictionless
pin at the end of a thin extension (of negligi
ble mass) a distance 7 m from the center of
mass of the rod.
Initially the rod makes an
angle of 54
◦
with the horizontal. The rod is
released from rest at an angle of 54
◦
with the
horizontal, as shown in the figure.
7 m
7 m
2
.
7 kg
O
54
◦
What is the angular speed of the rod at
the instant the rod is in a horizontal position?
The acceleration of gravity is 9
.
8 m
/
s
2
and the
moment of inertia of the rod about its center
of mass is
I
cm
=
1
12
m ℓ
2
.
Correct answer: 1
.
44603 rad
/
s.
Explanation:
Let :
ℓ
= 7 m
,
θ
= 54
◦
,
and
m
= 2
.
7 kg
.
Rotational kinetic energy is
K
R
=
1
2
I ω
2
and gravitational kinetic energy is
K
trans
=
m g d .
The inertia of the system is
I
=
I
cm
+
m d
2
=
1
12
m ℓ
2
+
m ℓ
2
=
13
12
m ℓ
2
.
Since the rod is uniform, its center of mass
is located a distance
ℓ
from the pivot.
The
vertical height of the center of mass above
the horizontal is
ℓ
sin
θ .
Using conservation of
energy,
K
i
+
U
i
=
K
f
+
U
f
K
f
=
U
i
1
2
I ω
2
=
m g ℓ
sin
θ
13
24
m ℓ
2
ω
2
=
m g ℓ
sin
θ
ω
2
=
24
13
g
sin
θ
ℓ
ω
=
radicalbigg
24
g
sin
θ
13
ℓ
=
radicalBigg
24 (9
.
8 m
/
s
2
) sin 54
◦
13 (7 m)
=
1
.
44603 rad
/
s
.
keywords:
002
10.0 points
A circularshaped object of mass 11 kg has
an inner radius of 15 cm and an outer radius
of 28 cm. Three forces (acting perpendicular
to the axis of rotation) of magnitudes 12 N,
24 N, and 13 N act on the object, as shown.
The force of magnitude 24 N acts 20
◦
below
the horizontal.
12 N
13 N
24 N
20
◦
ω
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alexander (jra2623) – homework 21 – Turner – (92510)
2
Find the magnitude of the net torque on
the wheel about the axle through the center
of the object.
Correct answer: 3
.
4 N
·
m.
Explanation:
Let :
a
= 15 cm = 0
.
15 m
,
b
= 28 cm = 0
.
28 m
,
F
1
= 12 N
,
F
2
= 24 N
,
F
3
= 13 N
,
and
θ
= 20
◦
.
F
1
F
3
F
2
θ
ω
The total torque is
τ
=
a F
2

b F
1

b F
3
= (0
.
15 m) (24 N)

(0
.
28 m) (12 N + 13 N)
=

3
.
4 N
·
m
,
with a magnitude of
3
.
4 N
·
m
.
003
10.0 points
The hour and minute hands of a tower clock
like Big Ben in London are 2
.
72 m and 4
.
58 m
long and have masses of 52
.
4 kg and 110 kg,
respectively.
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 Summer '08
 Kaplunovsky
 Mass, Work, Moment Of Inertia, Rigid Body, Correct Answer, Alexander

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