Rotations - Biomechanics of the Human Body - page 58

it will continue at rest, and if it is moving at constant velocity, it will continue
moving at constant velocity only if a resultant force F that we will call here simply
force F (Newton's first law, already presented in Chap. 1 ) is not applied on
it. On the contrary, if a force F is applied on the body, it will be accelerated with
acceleration a

t , and the larger the magnitude of force F , the larger will be
the acceleration acquired by the body (Newton's second law, already presented in
Chap. 1 ). If we double the force applied on this body, the acceleration will double
and, with triple the force, the acceleration is also tripled and so on. We can write
mathematically ( 4.1 ) that F / a is constant for a given body. This constant is
characteristic for a given body and is its mass m :

¼ Δ v /

Δ

F
a ¼

m : (4.1)

This equation indicates that for a given applied force F , the larger the body
mass, the less will be the acceleration acquired by it. Therefore, the mass of the
body is a measure of body inertia, with obvious relevance for the difficulty in
accelerating as well as decelerating an object. The jabiru is a bird found in the
Pantanal of Mato Grosso, Brazil, which has a large body mass and to take off needs
to run a lot to acquire enough velocity, and lands awkwardly, unlike other flying
birds with less mass that move gracefully. Body inertia is directly proportional to its
mass. Moreover, for translational motions, how the body mass is distributed is not
important.
In rotational motions of a body about an axis, its state of motion is only modified
if a resultant torque M acts on it. The difficulty here in changing its state of rotation
depends not only on body mass but on how it is distributed about an axis of rotation.
Equation ( 4.2 ) is equivalent to ( 4.1 ) for rotational motions:

M
α ¼

I ;

(4.2)

with M the magnitude of net torque, I the moment of inertia, and α the angular

acceleration.
The physical quantities of ( 4.1 ) and ( 4.2 ) that correspond are M

$ F , I $ m , and

α$ a . As the mass m is related to the body's inertia in translational motions, the

moment of inertia I is related to rotational inertia, which implies the difficulty in
increasing or decreasing the angular acceleration. The moment of inertia increases
with the increase of both mass and distance that characterizes the distribution of this
mass about an axis of rotation, and hence, it will be larger as this distance increases.
Therefore, the shape as well as the mass of a body determines how difficult it is to
set it into rotation, as can be seen in Fig. 4.1 .
All three types of wheels in Fig. 4.1 have the same mass, but their shapes are
different, i.e., the mass composing them is differently arranged around the axis of
rotation. The mass that composes wheel (a) is farther from the rotation axis than in
wheel (b), and the wheel (b) mass further than wheel (c) mass. After a certain time