Chris
McWilliams
Chapter
10:
Rotation
of
a
Rigid
Object
About
a
Fixed
Axis
Practice
Problems
Problem
1:
A
wheel
rotates
with
a
constant
2
angular
acceleration
of
3.50
rad /s .
(A)
If
the
angular
speed
of
the
wheel
is
2.0
rad/s
at
t
=
0,
through
what
angular
displacement
does
the
wheel
rotate
in
2.00
s?
€
(B)
Through
how
many
revolutions
has
the
wheel
turned
during
this
time
interval?
(C)
What
is
the
angular
speed
of
the
wheel
at
t
=
2.00
s?
Solution:
(A):
1
Δθ = θ f − θ i = ω i t + αt 2
2
1
Δθ = (2.00rad /s)(2.00s) + (3.50rad /s2 )(2.00s) 2 2
Δθ = 11.0radians = 630°
(B):
€
 1rev 
Δθ = 630°
 = 1.75revolutions  360° 
(C):
€
ω f = ω i + αt = 2.00rad /s + (3.50rad /s2 )(2.00s)
ω f = 9.00rad /s
€
Problem
2:
Calculate
the
moment
of
inertia
of
a
uniform
rigid
rod
of
length
L
and
mass
M
about
an
axis
perpendicular
to
the
rod
and
passing
through
its
center
of
mass.
Chris
McWilliams
Chapter
10:
Rotation
of
a
Rigid
Object
About
a
Fixed
Axis
dx
Solution:
Express
dm
in
terms
of
dx:
dm = λdx =
Iy =
€
M
dx L
O
2
∫ r dm
L /2
M
M L /2 2
Iy = ∫ x
dx =
∫ x dx
L
L
−L / 2
−L / 2
2
3 L /2
Iy =
€
M x 
1
= ML2
 
L  3 −L / 2 12
Problem
3:
A
uniform
rod
of
length
L
and
mass
M
is
attached
at
one
end
to
a
frictionless
pivot
and
is
free
to
rotate
about
the
pivot
in
the
vertical
plane.
The
rod
is
released
from
rest
in
the
horizontal
position.
What
are
the
initial
angular
acceleration
of
the
rod
and
the
initial
translational
acceleration
of
its
right
end?
Solution:
 L
τ ext = Mg 
 2
τ Mg(L /2) 3g
α= =
=
1
I
2
2L
ML
3
3g
at = rα =
r
2L
r=l
3gL 3
at =
= g
2L 2
€
Mg