phy1020.DVI

8.4 The Conical Pendulum.

Aconical pendulumis also similar to a simple plane pendulum, except that the pendulum is constrained to
move along the surface of a cone, so that the massmmoves in a horizontal circle of radiusr, maintaining a
constant anglefrom the vertical.
For a conical pendulum, we might ask: what speedvmust the pendulum bob have in order to maintain
an anglefrom the vertical? To solve this problem, let the pendulum have lengthL, and let the bob have
massm. A general approach to solving problems involving circular motion like this is to identify the force
responsible for keeping the mass moving in a circle, then set that equal to the centripetal forcemv^2 =r.In
this case, the force keeping the mass moving in a circle is the horizontal component of the tensionT, which
isTsin. Setting that equal to the centripetal force, we have

TsinD

mv^2

r

: (8.9)

The vertical component of the tension is

TcosDmg (8.10)

Dividing Eq. (8.9) by Eq. (8.10),

tanD

v^2

gr

(8.11)

Figure 8.3: A torsional pendulum. (Ref. [1])

From geometry, the radiusr of the circle is
Lsin. Making this substitution, we have

tanD

v^2

gLsin

: (8.12)

Solving for the speedv, we finally get

vD

p

Lgsintan: (8.13)

8.5 The Torsional Pendulum

Atorsional pendulum(Fig. 8.3) consists of a mass
mattached to the end of a vertical wire. The body
is then rotated slightly and released; the body then
twists back and forth under the force of the twisting
wire. As described earlier, the motion is governed
by the rotational version of Hooke’s law, D
.

8.6 The Physical Pendulum

Aphysical pendulumconsists of an extended body
that allowed to swing back and forth around some
pivot point. If the pivot point is at the center of mass,
the body will not swing, so the pivot point should be
displaced from the center of mass. As an example,