2.8. MATRICES AND CALCULUS 73

Using 2.33, in 2.31,aB = a−Ω× (Ω×R)

−2ω [(−y′ cosϕ) i+(x′ cosϕ+ z′ sinϕ) j− (y′ sinϕ)k] .

Neglecting the small term, Ω× (Ω×R) , this becomes

= −gk+T/m−2ω [(−y′ cosϕ) i+(x′ cosϕ+ z′ sinϕ) j− (y′ sinϕ)k]

where T, the tension in the string of the pendulum, is directed towards the point at whichthe pendulum is supported, and m is the mass of the pendulum bob. The pendulum can bethought of as the position vector from (0, 0, l) to the surface of the sphere x2+y2+(z − l)

2=

l2. Therefore,

T = −T xli−T y

lj+T

l − z

lk

and consequently, the differential equations of relative motion are

x′′ = −T x

ml+ 2ωy′ cosϕ

y′′ = −T y

ml− 2ω (x′ cosϕ+ z′ sinϕ)

and

z′′ = Tl − z

ml− g + 2ωy′ sinϕ.

If the vibrations of the pendulum are small so that for practical purposes, z′′ = z = 0, thelast equation may be solved for T to get

gm− 2ωy′ sin (ϕ)m = T.

Therefore, the first two equations become

x′′ = − (gm− 2ωmy′ sinϕ)x

ml+ 2ωy′ cosϕ

andy′′ = − (gm− 2ωmy′ sinϕ)

y

ml− 2ω (x′ cosϕ+ z′ sinϕ) .

All terms of the form xy′ or y′y can be neglected because it is assumed x and y remainsmall. Also, the pendulum is assumed to be long with a heavy weight so that x′ and y′ arealso small. With these simplifying assumptions, the equations of motion become

x′′ + gx

l= 2ωy′ cosϕ

andy′′ + g

y

l= −2ωx′ cosϕ.

These equations are of the form

x′′ + a2x = by′, y′′ + a2y = −bx′ (2.34)

where a2 = gl and b = 2ω cosϕ. Then it is fairly tedious but routine to verify that for each

constant, c,

x = c sin

(bt

2

)sin

(√b2 + 4a2

2t

), y = c cos

(bt

2

)sin

(√b2 + 4a2

2t

)(2.35)