H103CircularExample04.doc;9/9/2004
補充例題 103-4
Figure 6.12a shows a conical pendulum. Its bob, whose mass m is 1.5kg, whirls around in a
horizontal circle at constant speed v at the end of a cord whose length L, measured to the
center of the bob, is 1.7m. The cord makes an angle θ of 37° with the vertical. As the bob
swings around in a circle, the cord sweeps out the surface of a cone. Find the period τ of the
pendulum bob.
Solution: Figure 6.12b, the free-body diagram for the bob, shows the forces on the bob: the
pull T from the cord due to the cord’s tension, and the bob’s weight W (＝mg). We plane the
origin of axes at the center of the bob, as shown. Instead of the usual x axis (which is
stationary), we use a radial axis r that always points from the bob toward the center of the
circle.
The y and r components of T are Tcosθand Tsinθ, respectively. Since a y = 0 ,
Newton’s second law gives
T cos θ − mg = ma y = 0 or T cos θ = mg
(6-21)
There must be a net force along the r axis to provide the centripetal acceleration for the bob.
The only force component in that direction is Tsinθ. So along the r axis, Eq.6-20 gives
mv 2
T sin θ = mar =
(6-22)
R
Where R is the radius of the bob’s circular path. Dividing Eq.6-22 by Eq.6-21 and solving for
v, we obtain
gR sin θ
v=
cos θ
We can substitute 2πR / τ (the distance around the circle divide by the period) for the speed
v of the bob. Doing so and solving for τ, we obtain
R cos θ
(6-23)
g sin θ
Now, from Fig.6-12a we see that R = L sin θ . Making this substitution in Eq.6-23 yields
L cos θ
(6-24)
τ = 2π
g
τ = 2π
= 2π
(1.7m)(cos 37°)
= 2.3s
9.8m / s 2
(Answer)
From Eq.6-24 we see that the period τ does not depend on the mass of the bob, but only on
L cos θ , the vertical distance of the bob from its point of support. Thus, as shown in
Fig.6-12c, if several conical pendulums of different lengths are swung from the same support
with the same period, their bobs will all lie in the same horizontal plane.
H103CircularExample04.doc;9/9/2004
FIGURE 6-12 Sample Problem 6-9. (a) A conical pendulum, its cord making an
angle θ with the vertical. (b) A free-body diagram for the pendulum bob. The axes
point in the vertical and radial directions. The resultant force (and thus the
acceleration) points radially inward toward the center of the circle. (c) There
pendulums, of different lengths, are whirled around by a rotating shaft; their bobs
circulate in the same horizontal plane, as Eq.6-24 predicts.
資料來源：Halliday, 5th edition, Sample Problem 6-9