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MEM238- Dynamics
Summer 2013
FINAL EXAM SOLUTIONS
(Closed Book, Closed Notes, Show All Work, 5 Problems)
Basic Formulas:
2 2 2
(Law of sines)
sin sin sin
c 2 cos (Law of cosines)
a b c
a b ab
-1 -1 -
2 2 2
For vector : cos , cos , cos ,
cos cos cos 1 (Relation between direction cosines)
cos (Dot product)
A
A B
y x z
A A A
A A A
AB
(CoefficientofRestitutio n)
( ) ( )
(PrincipleofAngularImpulseandMomentum)
( ) ( ) (PrincipleofLinearImpulseandMomentum)
(Power)
(WorkofaConstantCoupleMoment)
(WorkofaLinearElasticSpring)
(Principle ofWorkandEnergy)
cos (WorkofaForce)
, 2 (RadialandTransverseAcceleration)
, (RadialandTransverseVelocity)
, (TangentialandNormalAcceleration)
, 2 ( )(ConstantAcceleration-AngularMotion) 2
, 2 ( )(ConstantAcceleration-LinearMotion) 2
, , (DifferentialKinematicRelationships-Angular Motion)
, , (DifferentialKinematicRelationships-LinearMotion)
1 1
2 2
1 2
1 2
1 1 2 2
1 2
2
2 2 2
2 2 2
2
1
G
2
1
G
2
1
2
1
B
B
o o
t
o t
t
t
M
s
r
r
t n
o c o o c o c o
o c o o c o c o
v A
v
A
v v
e
dt
m dt m
dt
dU P
U M
ks ks s
U
T U T
U d F ds
a r r a r r
v rv r
v a va
r n
ra t
a
v r
t t t
v v ats s vt at v v a s s
d d dt
d
dt
d
ads vdv dt
ds v dt
dv a
s
H M H
v F v
Fv
F r
a αr r
v ωr
r
r
Problem 1
For the disk shown, 0 = 1rad/s and r = 0.5m. Determine (a)(9pts) the angular velocity
AB of link AB and (b)(9pts) the velocity v B of collar B for the configuration shown in the
figure. If the angular velocity 0 of the disk is constant, determine (c)(6pts) the angular
acceleration AB
of link AB.
Problem 3
An 8lb uniform rectangular door opens upward and is counterbalanced with a spring as
shown in the figure. The modulus and undeformed length of the spring are k = 24lb/ft and
l o = 23in., respectively. If the door has an angular velocity of 3rad/s counterclockwise
when it is vertical ( = 0
o
), determine the angular velocity of the door when it is
horizontal ( = 90
o
). Model the door as a uniform, slender rod. (20pts)
Problem 4
The horizontal velocity of the landing airplane is 50 m/s, its vertical velocity (rate of
descent) is 2 m/s, and its angular velocity is zero. The mass of the airplane is 12 Mg, and
the moment of inertia about its center of mass is 1 X 10
5
kg m
2
. Neglect the horizontal
force exerted on the wheels by the runway. (a)(3pts) Draw the impulse-momentum
diagram for the airplane. Determine (b)(10pts) the angular velocity of the airplane and
(c)(10pts) the linear velocity of its center of mass just after it touches down if its wheels
don’t stay in contact with the runway and the coefficient of restitution of the impact
is e = 0.4.
Problem 5
The large rotor has a mass of 60kg and a radius of gyration about its vertical axis of
200mm. The small rotor is a solid circular disk with a mass of 8kg and is initially rotating
with an angular velocity 1 = 80rad/s with the large rotor at rest. A spring-loaded pin P
which rotates with the large rotor is released and bears against the periphery of the small
disk, bringing it to a stop relative to the large rotor. Neglect any bearing friction and
calculate the final angular velocity of the assembly. (8pts)
