ac/The total force in the
forward-backward direction is
zero in both cases.
ad/There is no outward force
on the bowling ball, but in the
noninertial frame it seems like
one exists.
The magnitude of the acceleration is|a|=√
a^2 x+a^2 y
=ω^2 r.It makes sense thatωis squared, since reversing the sign ofω
corresponds to reversing the direction of motion, but the acceler-
ation is toward the center of the circle, regardless of whether the
motion is clockwise or counterclockwise. This result can also be
rewritten in the form
|a|=
|v|^2
r.
Although I’ve relegated the resultsa=ω^2 r=|v|^2 /rto an ex-
ample because they are a straightforward corollary of more general
principles already developed, they are important and useful enough
to record for later use. These results are counterintuitive as well.
Until Newton, physicists and laypeople alike had assumed that the
planets would need a force to push themforward in their orbits.
Figure ab may help to make it more plausible that only an inward
force is required. A forward force might be needed in order to cancel
out a backward force such as friction, ac, but the total force in the
forward-backward direction needs to be exactly zero for constant-
speed motion. When you are in a car undergoing circular motion,
there is also a strong illusion of anoutwardforce. But what object
could be making such a force? The car’s seat makes an inward force
on you, not an outward one. There is no object that could be ex-
erting an outward force on your body. In reality, this force is an
illusion that comes from our brain’s intuitive efforts to interpret the
situation within a noninertial frame of reference. As shown in figure
ad, we can describe everything perfectly well in an inertial frame
of reference, such as the frame attached to the sidewalk. In such a
frame, the bowling ball goes straight because there isnoforce on it.
The wall of the truck’s bed hits the ball, not the other way around.Integration
An integral is really just a sum of many infinitesimally small
terms. Since vector addition is defined in terms of addition of the
components, an integral of a vector quantity is found by doing inte-
grals component by component.
Projectile motion example 73
.Find the motion of an object whose acceleration vector is con-
stant, for instance a projectile moving under the influence of grav-
ity.
.We integrate the acceleration to get the velocity, and then inte-
grate the velocity to get the position as a function of time. Doing214 Chapter 3 Conservation of Momentum