Mathematics of Physics and Engineering

42 Kinematics and Dynamics of a Point Mass

L — mr x r = m(£r x 196) = m£^29 A, and

^=m£r6k. (2.1.7)
at
The forces acting on the pendulum are the weight W of m, the air resistance
Fa and the force Fp exerted by the pin at O. Clearly, W = mgi, where
g is the acceleration of gravity. Physical considerations suggest that the
force Fa on m may be assumed to act tangentially to the circular path and
to be proportional to the tangential velocity: Fa = —c£9 6, where c is the
damping constant. The total torque T about O exerted by these forces is

T = r xW + r x Fa + Ox Fp, = £rxmgi-£rxc£9 8,

or

T=-(mg£ sin 9 + £^2 C9)K. (2.1.8)

Since the frame O is inertial, equation (2.1.6) applies, and by (2.1.7) and

(2.1.8) above, we obtain ml? 6 = —mg£sin6 — l^2 c6, or

m£e + d6 = -mg sin9. (2.1.9)

Equation (2.1.9) is a nonlinear ordinary differential equation and cannot
be integrated in quadratures, that is, its solution cannot be written using
only elementary functions and their anti-derivatives. When c = 0, such a
solution does exist and involves elliptic integrals; see Problem 2.3, page 417,
if you are curious.
The more familiar harmonic oscillator

6 = -{g/£)6 (2.1.10)

is obtained from (2.1.9) when c = 0 and 6 is small so that sin0 w 6; this
equation should be familiar from the basic course in ordinary differential

equations. If 9(0) = (^0) O and 0(0) = 0, then the solution of (2.1.10) is
0(t) = (^0) O cos(cjt), where w = (£/g)1/2.
The period of the small undamped oscillations is 2n(£/g)^1 /^2 , and the value
of £ can be adjusted to provide a desired ticking rate for a clock mechanism.
The idea to use a pendulum for time-keeping was studied by the Italian
scientist GALILEO GALILEI (1564-1642) during the last years of his life,
but it was only in 1656 that the Dutch scientist CHRISTIAAN HUYGENS
(1629-1695) patented the first pendulum clock.