Rotations - Biomechanics of the Human Body - page 69

Fig. 4.10 Representation
of a ballerina who is able to
increase her rotational
velocity by a maneuver of
closing her arms, as the
rotational angular
momentum has to be
conserved

4.5.1 Angular Impulse

Now we introduce another physical quantity: the angular impulse ( 4.11 ). It is given
by the product of external net torque and the duration of the action and is responsi-
ble for the variation of angular momentum,

L ¼ L final L initial , of a body:

Δ

Angular impulse ¼ Torque ext Δ t ¼ Δ L : (4.11)

The rotational motions of the human body occur about the axis that passes
through its center of gravity. In this case, an important observation is that the
weight force acting on the body's center of gravity does not produce torque and,
hence, does not change its angular momentum. Torques are originated by impulse
forces that will introduce or change the angular momentum of body. In case that the
impulse force is not applied, the body will maintain its state of rotation, that is, its
angular momentum is conserved.
Figure 4.10 shows a ballerina in evolution. She is able to increase her rotational
velocity simply by closing her arms. When she is with outstretched arms, she has
a greater difficulty to rotate, i.e., she has more rotational inertia (moment of inertia)
than with closed arms. Therefore, since the angular momentum is conserved,
when she closes her arms, an increase in the angular velocity must occur, according
to ( 4.10 ).

Example 4.4 Consider an ice skater with 60 kg mass and radius of gyration of
0.15 or 0.11 m about the main longitudinal axis when she is with opened or
closed arms, respectively. Considering that her angular velocity is 6 rad/s with
opened arms, determine her angular velocity when she closes her arms. Calculate
the number of turns, supposing that she remains with this angular velocity (closed
arms) during 30 s.

mk 2 (60 kg)(0.15 m) 2 (6 s 1 ) 8.1 kg m 2 /s. Opened arms: L ¼ I ω ¼ ω ¼ ¼

Closed arms: the angular momentum is conserved after her maneuver. Hence,
8.1 kg m 2 /s (60 kg)(0.11 m) 2 ¼ ω :