Circular Motion and Other Applications of Newton's Laws

162 CHAPTER 6 Circular Motion and Other Applications of Newton’s Laws
EXAMPLE 6.9
Fictitious Forces in Linear Motion
A small sphere of mass m is hung by a cord from the ceiling
of a boxcar that is accelerating to the right, as shown in Figure
6.13. According to the inertial observer at rest (Fig.
6.13a), the forces on the sphere are the force T exerted by
the cord and the force of gravity. The inertial observer concludes
that the acceleration of the sphere is the same as that
of the boxcar and that this acceleration is provided by the
horizontal component of T. Also, the vertical component of
T balances the force of gravity. Therefore, she writes Newton’s
second law as �F � T � mg � ma, which in component
form becomes
�
(1) �Fx � T sin � � ma
Inertial observer
(2) �Fy � T cos � � mg � 0
Thus, by solving (1) and (2) simultaneously for a, the inertial
observer can determine the magnitude of the car’s acceleration
through the relationship
a � g tan �
mg
T
T
F fictitious
θ
θ
mg
(a)
(b)
Because the deflection of the cord from the vertical serves as
a measure of acceleration, a simple pendulum can be used as an
accelerometer.
According to the noninertial observer riding in the car
(Fig. 6.13b), the cord still makes an angle � with the vertical;
however, to her the sphere is at rest and so its acceleration is
zero. Therefore, she introduces a fictitious force to balance
the horizontal component of T and claims that the net force
on the sphere is zero! In this noninertial frame of reference,
Newton’s second law in component form yields
Noninertial observer
� �F� x � T sin � � F fictitious � 0
�F� y � T cos � � mg � 0
If we recognize that F fictitious � ma inertial � ma, then these expressions
are equivalent to (1) and (2); therefore, the noninertial
observer obtains the same mathematical results as the inertial
observer does. However, the physical interpretation of the
deflection of the cord differs in the two frames of reference.
a
Noninertial
observer
Inertial
observer
Figure 6.13 A small sphere suspended from the ceiling of a boxcar accelerating to the right is deflected
as shown. (a) An inertial observer at rest outside the car claims that the acceleration of the
sphere is provided by the horizontal component of T. (b) A noninertial observer riding in the car says
that the net force on the sphere is zero and that the deflection of the cord from the vertical is due to a
fictitious force F fictitious that balances the horizontal component of T.