Biomedical Engineering Reference
In-Depth Information
2.3
Newton's Laws
The concepts in this biomechanics textbook are based on the work of Sir Isaac
Newton (1643-1727). In his most famous work 'Philosophiae Naturalis Principia
Mathematica' he described the law of gravity and what are currently known as the
three laws of Newton, forming the basis for classical mechanics. These laws are:
•
Every object in a state of uniform motion tends to remain in that state of motion unless
an external force is applied to it. This is often termed simply the 'Law of Inertia'.
•
In a one-dimensional context the second law states that the force
F
on an object equals
the mass
m
, with SI unit [kg], of the object multiplied by the acceleration
a
, with
dimension [m s
−
2
], of the object:
F
=
ma
.
(2.5)
Consequently, the force
F
has the dimension [N] (Newton), with
1[N]
=
1[kgms
−
2
].
This may be generalized to the three-dimensional space in a straightforward manner.
Let the position of a material particle in space be given by the vector
x
. If the particle
moves in space, this vector will be a function of the time
t
, i.e.
x
=
x
(
t
).
(2.6)
The velocity
v
of the particle is given by
dx
dt
,
v
(
t
)
=
(2.7)
and the acceleration
a
follows from
d
2
d
v
dt
=
x
dt
2
.
a
(
t
)
=
(2.8)
Newton's second law may now be formulated as
F
=
ma
.
(2.9)
•
The third law states that for every action there is an equal and opposite reaction.
This law is exemplified by what happens when we step off a boat onto the bank of
a lake: if we move in the direction of the shore, the boat tends to move in the opposite
direction.
Example 2.2
Let the position of a particle with mass
m
for
t
≥
0 be given by
1
2
t
τ
x
(
t
)
=
+
x
0
,
