399 Problems
z
(0) = 0, η(0) = 1. (a) Use Euler’s dynamical equations and the (˙, ˙η) kinematical
equations to obtain numerical values of ω
x
,ω
y
,ω
z
,
x
,
y
,
z
,η by integrating over
the time interval 0 ≤ t ≤ 1 s. For times within this interval, calculate the work W
done by M
z
and also the angular deviation α of the x-axis from its original orientation,
where α = cos
−1
(1 −2
2
y
− 2
2
z
). (b) Find W
max
and the time at which it occurs. (c)
Determine α
max
and the time of occurrence. (d) Find the time at which ω
x
,ω
y
,ω
z
simultaneously return to their original values.
6.10. A thin uniform disk of mass m and radius r rolls without slipping on a horizontal
surface. The generalized coordinates are (ψ, θ, φ, X, Y ) where (ψ, θ, φ) are type
I Euler angles, and (X, Y ) is the location of the contact point. The contact point
has a velocity r
˙
φ at an angle ψ with the positive X -axis, that is,
˙
X = r
˙
φ cos ψ
and
˙
Y =−r
˙
φ sin ψ. Choose (
˙
ψ,
˙
θ,)asus where =
˙
φ −
˙
ψ sin θ is the angu-
lar velocity of the disk about its symmetry axis. Assume that g/r = 20 s
−2
, where
g = 9.81 m/s
2
and r = 0.4905 m. (a) Derive the three differential equations of mo-
tion, that is, equations for
˙
usintermsofqs and us. (b) Assume the initial conditions
ψ(0) = 0, θ (0) = π/6, φ(0) = 0, X (0) = 0, Y (0) = 0,
˙
ψ(0) = 4 rad/s,
˙
θ(0) = 0,
(0) = 2 rad/s. Use numerical integration to obtain the values of the qs and us over
the interval 0 ≤ t ≤ 3.5s.Findθ
min
and the following θ
max
with their corresponding
times, as well as the times when θ = 0. Show that the motion of θ is periodic, but the
path of the contact point on the XY-plane is not periodic.
mg
r
i
j
(X, Y )
disk
θ
⭈
.
.
Figure P 6.10.
6.11. A dumbbell consists of two particles, each of mass m, connected by a massless rod of
length l (Fig. P 6.11). Given that m = 1 kg, l = 1 m, and g = 9.8m/s
2
. Choose (x,θ)
as generalized coordinates and assume the initial conditions x(0) =
1
2
m,
˙
x(0) = 0,
θ(0) = 0,
˙
θ(0) =
1
2
rad/s. As the dumbbell falls due to gravity and slides without
friction, particle A hits the wall inelastically. Then it slides down the wall until it
finally leaves and moves to the right. Use the notation v ≡
˙
x, ω ≡
˙
θ. Consider three
phases of the motion.