Simple Nature - Light and Matter

ab/This figure shows an in-

tuitive justification for the fact

proved mathematically in the

example, that the direction of the

force and acceleration in circular

motion is inward. The heptagon,

2, is a better approximation to

a circle than the triangle, 1. To

make an infinitely good approx-

imation to circular motion, we

would need to use an infinitely

large number of infinitesimal taps,

which would amount to a steady

inward force.

or

dr

dt

=

dx

dt

xˆ+

dy

dt

yˆ+

dz

dt

ˆz.

All of this reasoning applies equally well to any derivative of a vector,
so for instance we can take the second derivative,

ax=

dvx

dt

, ay=

dvy

dt

, az=

dvz

dt

or

dv

dt

=

dvx

dt

xˆ+

dvy

dt

yˆ+

dvz

dt

ˆz.

A counterintuitive consequence of this is that the acceleration
vector does not need to be in the same direction as the motion. The
velocity vector points in the direction of motion, but by Newton’s
second law,a=F/m, the acceleration vector points in the same di-
rection as the force, not the motion. This is easiest to understand if
we take velocity vectors from two different moments in the motion,
and visualize subtracting them graphically to make a ∆vvector.
The direction of the ∆vvector tells us the direction of the accelera-
tion vector as well, since the derivative dv/dtcan be approximated
as ∆v/∆t. As shown in figure aa/1, a change in the magnitude of
the velocity vector implies an acceleration that is in the direction of
motion. A change in the direction of the velocity vector produces
an acceleration perpendicular to the motion, aa/2.
Circular motion example 72
.An object moving in a circle of radiusrin thex-yplane has

x=rcosωt and

y=rsinωt,

whereωis the number of radians traveled per second, and the

positive or negative sign indicates whether the motion is clock-

wise or counterclockwise. What is its acceleration?

.The components of the velocity are

vx=−ωrsinωt and

vy= ωrcosωt,

and for the acceleration we have

ax=−ω^2 rcosωt and

ay=−ω^2 rsinωt.

The acceleration vector has cosines and sines in the same places

as thervector, but with minus signs in front, so it points in the op-

posite direction, i.e., toward the center of the circle. By Newton’s

second law,a=F/m, this shows that the force must be inward as

well; without this force, the object would fly off straight.

Section 3.4 Motion in three dimensions 213