Dynamics: Practice Final – 3-D Kinematics of Rigid Bodies
Show all work for partial credit. Use additional paper if necessary.
1.
Find the components of the inertia matrix for the compound bar
shown. The bar has a mass of 3 kg/m. (A slender bar has a
moments of inertia about its center 1/12 m·l2, its moment of inertia
about its long axis is zero, and its products of inertia about its
center are zero).
Ixx = ___________________ kg·m2
Izz = ___________________ kg·m2
Iyz = ___________________ kg·m2
2.
A certain rigid body has an inertia matrix of [I], given below. What is the body’s moment of inertia if it
is spun about an axis in the direction û , also given below?
⎡ 20 0 − 5⎤
⎡3 / 5 ⎤
⎥
⎢
2
[I]= ⎢ 0 15 − 5⎥ kg·m
û = ⎢⎢ 0 ⎥⎥
⎢⎣− 5 − 5 10 ⎥⎦
⎢⎣4 / 5⎥⎦
3.
A certain rigid body has an inertia matrix of [I], given below. What is the body’s angular momentum if
r
it is spun with angular velocity ω , also given below?
⎡ 20 0 − 5⎤
⎡1⎤
r ⎢⎥
⎥
⎢
2
ω = ⎢1⎥ rad/sec
[I]= ⎢ 0 15 − 5⎥ kg·m
⎢⎣− 5 − 5 10 ⎥⎦
⎢⎣1⎥⎦
4.
A certain rigid body has an inertia matrix of [I], given below. What is the body’s rotational kinetic
r
energy if it is spun with angular velocity ω , also given below?
⎡ 20 0 − 5⎤
⎡1⎤
r ⎢⎥
⎢
⎥
2
ω = ⎢1⎥ rad/sec
[I]= ⎢ 0 15 − 5⎥ kg·m
⎢⎣− 5 − 5 10 ⎥⎦
⎢⎣1⎥⎦
5.
A uniform 40 kg, 2 meter rod floating in space and aligned along the z-axis experiences two forces
applied at its ends. Force A is 30 N in the x-direction, and force B is 20 N in the y-direction. Find the
following values at this instant:
• The inertial linear acceleration of the center of mass
• The angular acceleration about the center of mass
MAE-2102
1
Ver. D, Rev. 1
D
Dynamic
cs: Practicce Final – 3-D Kinnematics of
o Rigid Bodies
B
6.
Att the instant sho
own, the towerr crane is rotatiing about the z axis
with an angular velocity
v
ω1 = 0.25
0 rad/s, whicch is increasingg at
2
0.66 rad/s . The boom
b
OA is rottating downwarrd with an anguular
vellocity ω2 = 0.4
4 rad/s, which is increasing att 0.8 rad/s2.
•
•
•
ω&1 =0.6 rad/s2
B
Determine th
he angular veloocity of the booom
Calculate thee velocity of pooint B located at the top of thhe
boom.
Find the acceleration of pooint B.
ω& 2 =0.8 rad/s2
7.
own, the base of
o the robotic arm
a is turning
Att the instant sho
aboout the z axis with
w an angularr velocity of ω1 =4 rad/s,
whhich is increasing at ω&1 = 3 raad/s2. Also, thee boom
seggment BC is ro
otating at
ω BC = 8 rad/s, whicch is increasingg
vC=1 m/s
at ω& BC = 2 rad/s2. The gripper is retracting att a constant ratte
vC =1 m/s.
•
•
C
Caalculate the velocity of point C.
(exxtra) Find the acceleration
a
off point C.
8.
A 4 kg thin ring of
o diameter 1 meter
m
hangs froom a nail in a wall.
w
If the rinng is started
roccking in the plaane of the ring about the nail with a small angular
a
displaceement, what is the
perriod of oscillattion of the ringg?
E-2102
MAE
2
Ver. D, Rev. 1