184
4 Rotational Dynamics
Fig. 4.33
around the ring in the counterclockwise sense. The only forces acting in a
horizontal plane are the reactions at P which are equal and opposite. Consequently G will not move and the angular momentum about G which was zero
initially will remain zero throughout the motion due to its conservation.
m · PG(v − PGω) − I G ω = 0
( 1 )
where ω is the angular velocity of the ring.
Now PG =
Mr
M + m
, OG =
mr
M + m
(2)
I ω = I CM + M(OG)
2
= Mr
2
+ M
m 2 r 2
(M + m) 2
(3)
Using (2) and (3) in (1) and simplifying we obtain
ω =
mv
(M + 2m)r
(4)
4.3.3 Coriolis Acceleration
4.63 (a) ω points in the south to north direction along the rotational axis of the
earth.
ω =
2π
T
=
2π
86, 160
= 7.292 × 10
−5 rad/s
(b) The period of rotation of the plane of oscillation is given by
T
=
2π
ω =
2π
ω sin λ
=
T 0
sin λ
=
24
sin 30 ◦ = 48 h
4 Rotational Dynamics
Fig. 4.33
around the ring in the counterclockwise sense. The only forces acting in a
horizontal plane are the reactions at P which are equal and opposite. Consequently G will not move and the angular momentum about G which was zero
initially will remain zero throughout the motion due to its conservation.
m · PG(v − PGω) − I G ω = 0
( 1 )
where ω is the angular velocity of the ring.
Now PG =
Mr
M + m
, OG =
mr
M + m
(2)
I ω = I CM + M(OG)
2
= Mr
2
+ M
m 2 r 2
(M + m) 2
(3)
Using (2) and (3) in (1) and simplifying we obtain
ω =
mv
(M + 2m)r
(4)
4.3.3 Coriolis Acceleration
4.63 (a) ω points in the south to north direction along the rotational axis of the
earth.
ω =
2π
T
=
2π
86, 160
= 7.292 × 10
−5 rad/s
(b) The period of rotation of the plane of oscillation is given by
T
=
2π
ω =
2π
ω sin λ
=
T 0
sin λ
=
24
sin 30 ◦ = 48 h