Final for Dynamics (MAE/CEE 80)
Summer 2008
Duration: 2 Hours
Please make sure you do all your work on this exam (including the blank pages
provided).
Name:___________________________, ID#:___________________
A couple of Hints:
1- The moment of inertia of an object with mass m and radius of gyration k is
equal to I = mk 2 .
2- It would be best to keep symbols like g, L, IG, etc. as long as possible, and
plug the numerical values at the end.
3- Assume g=10 m/s2
Problem 1:__________
Problem 2:__________
Problem 3:__________
Problem 4:__________
Problem 5:__________
Total:________
Problem 1 (15 points) – Link AB has a constant counterclockwise angular velocity
ω AB = 5 rad / sec . The following problem applies to the incident shown below when
θ = 60 deg and links AB and CD are horizontal.
Clearly state the sign convention you choose for angular velocity.
7m
a) Find the velocity of point B(i.e., V B = ?) .
A
ω AB = 5
B
rad
s
10 m
θ=60˚
C
D
5m
b) Find the angular velocity of link BC and angular velocity of link CD. (i.e., ω BC = ? & ω CD = ? ) .
See the next page
for part (c)
c) Find the velocity of point C. (i.e., V C = ?) .
Problem 2 (18 points) – In the figure below, disc B is spinning about its center with a
constant angular velocity ω1 . Then the whole arm A starts rotating with a constant angular
velocity ω 2 , as shown. The radius of disc is R and length of arm A is L .
Note: The xyz coordinate system as shown, is global and fixed. If you wish to use different
coordinate system, you MUST identify it and state how it changes with time.
State your answers in terms of ω1 , ω 2 , R, L
Z
Determine
a) The total angular velocity of the disc B as the
whole assembly rotates (i.e., Ω B = ? ). Find
Ω B at the shown instant.
L
D
b) The total angular acceleration of the disc.
Y
(i.e., α B = ? ). Find α B at the shown instant.
c) Find the velocity of the point D on the rim, at
the instant shown.
X
Problem 3 (27 points)- The illustrated system is a spring-restrained cylinder that’s
connected to block A through an inextensible massless string. The string goes over another
cylindrical body that rotates as the string moves (no slip between the body and the
string). Cylinder B also rolls without slipping. Both cylindrical bodies have a radius of
0.5m, a mass of 4kg, and a moment of inertia of 1 kg.m2. Mass of block A is equal to 10kg
and the coefficient of the spring is k=78 N/m. The system is released from rest with the
spring in its unscratched length.
(a) What is the relationship between displacement
of the cylinder B and the block A.
(Pay attention to the constraint).
B
C
(b) Obtain the velocity ( v B ) and angular velocity of cylinder B ( ω B ) in terms of velocity of
block A ( v A ).
(c) Obtain the angular velocity ( ω C ) of the cylinder C in terms of velocity of block A ( v A ).
See next page for part (d)
(d) Find the velocity of block A after it has fallen 2 m.
Problem 4 (15 points)- A 40-g bullet is fired with horizontal velocity of 205 m/s into the
middle of the slender 4-kg bar of length L=300 mm hanged vertically from point O.
Knowing that the bar is initially at rest, determine the angular velocity of the bar
immediately after the bullet becomes embedded.
NOTE: You need to show your work (e.g., freebody diagrams, setting up impulse
momentum relationship, etc.)
o
L
2
L
2
Problem 5 (25 points) – A simplified representation of moving
bicycle is shown below, where M=335 N.m is the constant torque
applied by the cyclist to the front chain ring.
The mass of the front chain ring and its radius of gyration are
mA=0.25 Kg and k A = 0.08 m. The total mass of the rear cog and
the wheel is mB=1 kg and their radius of gyration is k B = 0.35 m.
Note that the rear cog is attached to the rear wheel
A
Ay
r2 = 0.05 m
M
r1 = 0.1 m
Ax
R = 0 .4 m
B
N
µs = 0.8
Ignore the mass of the chain. The rear wheel rotates without slipping. The total weight of the
cyclist and the bike is 600 Newton and the rear tire supports 85% of it (N?). Assume the bike is
operating at the maximum friction condition. Add the remaining external forces to the figure above.
(a) What is the relationship between the angular acceleration of the front ring (α A ) and the angular
acceleration of the rear wheel (α B ) ?
(b) Find the angular acceleration of the rear wheel ( α B =?).
Hint: Consider the front ring and the rear wheel/cog separately. For each case, draw your
freebody diagrams (both force/moment and accelerations).