Environmental Engineering Reference
In-Depth Information
The first two terms in Eq. (1.26) are zero for uniform circular motion, so we obtain
rω
2
r
.
a
(
uniform circular motion
)
=−
(1.27)
Notice that the acceleration here is not a result of a change in the magnitude of
v
; this is constant. Rather, the direction of
ˆ
(and hence that of
v
) is constantly
changing and this gives rise to the acceleration in the radial direction. Notice also
that the acceleration in Eq. (1.27) points towards the centre of the circular orbit,
i.e. in the direction of
θ
r
.
We have derived Eq. (1.27) using the formal differentiation of the time-dependent
position vector
r
r
. We can also understand the result geometrically. We begin by
sketching the important vectors in Figure 1.9. We show the position of the particle
at times
t
and
t
−
+
t
(points
A
and
B
respectively) as well as the corresponding
velocity vectors. The velocity vectors are tangential to the path of the particle
and have equal magnitudes (
v
=
rω
). Let's construct the velocity difference
v
=
v
(t
v
(t)
: you should be able to see from the diagram that
v
points
approximately towards the centre of the circle from the midpoint of the circular arc
between
A
and
B
. In the triangle of velocity vectors formed by
v
(t
+
t)
−
t),
v
(t)
and
v
we can approximate the magnitude of
v
by a circular arc, and write
v
+
rω
2
t
. In the limit
t
≈
vθ
=
vωt
=
→
0 the approximation becomes
rω
2
, and the acceleration points exactly in the direction
exact,
a
=
v/t
→
−
r
.
We are therefore led to Eq. (1.27).
v
(
t
+ ∆
t
) =
v
(
t
) + ∆
v
v
(
t
)
∆
v
B
w∆
t
v
(
t
+ ∆
t
)
v
(
t
)
A
Figure 1.9 Uniform circular motion. Notice that the changing direction of the velocity
vector results in a vector
v
that points approximately towards the centre of the circle.
In the limit of vanishingly-small
t
this vector corresponds to the acceleration and points
exactly towards the centre.
1.4 STANDARDS AND UNITS
In this chapter we have introduced the concepts of space and time without saying
too much about measurement. Measurement of a physical quantity consists of
making a comparison of that quantity, either directly or indirectly, with a standard.
A standard is something on which we must all be able to agree and which defines
the unit in which the measurement will be expressed. We will illustrate the idea
