1.053J/2.003J Dynamics and Control I
Fall 2006
Final Exam
20
th
December, 2006
Important Notes:
1.
You are allowed to use three lettersize sheets (twosides) of notes.
2.
There are five problems on the exam for a total of 100 points.
The points
awarded for each problem are indicated at the top.
3.
You have 180 minutes.
Cite as: Sanjay Sarma, Nicholas Makris, Yahya ModarresSadeghi, and Peter So, course materials for 2.003J/1.053J
Dynamics and Control I, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology.
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Problem 1: Nonuniform disc (20 points)
Figure 1
A nonuniform disc of mass
m
and radius
r
has its center of mass
C
at a distance
d
from
the center of the disc
Q
, as shown in figure 1. The disc rolls on the horizontal surface
without slippage.
Initially, the disc is oriented such that
C
lies vertically below
Q
, after which it is
perturbed and starts rolling. Let
θ
be the instantaneous angle that the segment
QC
makes
with the vertical direction, measured in the anticlockwise direction as shown in the
figure, and let
I
be the moment of inertia of the disc about point
C.
Note that gravity acts.
a.
In terms of
θ
and its derivatives
θ
and
, find the acceleration of
C
with respect to a
frame that is fixed to the ground.
Now use the Newtonian method (linear and angular momentum equations) to derive the
equation of motion for the system as follows.
b.
Can you use the torque equation about the instantaneous center of rotation (ICR)?
Explain.
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 Spring '05
 GeorgeKocur
 Force, Nicholas Makris, Sanjay Sarma, Yahya ModarresSadeghi

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