7.3 Solutions
333
For θ = −
π
2
, (10) reduces to
k(a
2
− b
2
) + mgb
(12)
Expression (12) will be positive if a 2 > b 2 −
mgb
k
, and θ = −
π
2
will be
a stable point.
(iv) T = 2π
A(θ )
V (θ )
V
−
π
2
= k(a
2
− b
2
) + mgb, by (12)
A(θ ) is the coefficient of
1
2
˙
θ
2 in (3)
∴ A(θ ) = m(a
2 sin
2
θ + b
2 cos
2
θ)
∴ A
−
π
2
= ma
2
∴ T = 2π
ma 2
k(a 2 − b 2 ) + mgb
7.31 In prob. (7.12) the following equations were obtained:
(m 1 + m 2 ) l 1 ¨
θ 1 + m 2 l 2 ¨
θ 2 + (m 1 + m 2 )gθ 1 = 0
( 1 )
l 2 ¨
θ 2 + gθ 2 + l 1 ¨
θ 1 = 0
( 2 )
For l 1 = l 2 = l and m 1 = m 2 = m, (1) and (2) become
2l ¨
θ 1 + l ¨
θ 2 + 2gθ 1 = 0
( 3 )
l ¨
θ 2 + l ¨
θ 1 + gθ 2 = 0
( 4 )
The harmonic solutions of (3) and (4) are written as
θ 1 = A sin ωt, θ 2 = B sin ωt
( 5 )
¨
θ 1 = −Aω
2 sin ωt, ¨
θ 2 = −Bω
2 sin ωt
( 6 )
Substituting (5) and (6) in (3) and (4) and simplifying
2(lω
2
− g)A + lω
2 B = 0
( 7 )
lω
2 A + (lω
2
− g)B = 0
( 8 )
The frequency equation is obtained by equating to zero the determinant
formed by the coefficients of A and B: