2.003 Problem Set 7, Due October 28, 2010
1.
The recent quiz revealed some gaps in understanding of kinematics involving velocities and
acceleration on translating and rotating objects. This problem is meant to address that and to
give you some guided practice at solving kinematics problems. The exercise is to read the
introductory pages in the 1st reading from Williams, pp 75-109, which you can download from
Stellar. Then do yourself 2 or 3 of the examples in the reading.
a. Study examples 3.5 and 3.7
b. Do examples 3.8 and 3.9 and if you have time and motivation do 3.10 or 3.11.
By “do” we mean that you should work completely through the problem using the
notation you are used to using. Do all the problems using the 3D vector versions of the
velocity and acceleration formulas. The principal purpose of the exercise is to get you
more comfortable with computing V and A using rotating xyz frames.
2. This is a problem intended to give you some practice in using the mass moment of inertia
matrix, dP/dt and dH/dt, in finding equations of motion. It will also illustrate the most common
kind of rotational imbalance. This is long in words, but the individual steps are short.
a. Consider a uniform wheel, shown in figure A. Identify the principal axes passing through
G the center of mass. Let the z axis coincide with the axis of rotation through G. Write
out the complete matrix expression for [I/G], the inertia matrix with respect to G.
b. Let omega=0i+0j+omega-z-k. Compute H/G from the matrix product of [I/G] and the
omega vector. Find the time derivative of H/G = torque to obtain equations of motion
for the wheel. Show that there are no torques about the y or x axes. In other words
the wheel is dynamically balanced.
c. Find the linear moment P and dP/dt for the above. This is trivial seeming result, but
important to the next step.
d. In figure B, the wheel is shown with a small hunk of mass removed on the x axis at r =R
and z=0. The amount removed is m=0.001M, where M is the original mass of the
wheel. Find a new [I’/G] where G is still the original mass center and is the point
coinciding with the axis of rotation. Find the new I’zz term. The I’xx term is unaffected.
The I’yy term does change, but because the rotation is only about the z axis it does not
change the angular momentum, so you do not need to compute it. The rotation rate
vector is the same as before. Hint: I’zz+.001MR^2=I /Gzz.
e. Compute H’/G, and d(H’/G)/dt. Are there any unbalanced torques about the y and x
axes. Is the wheel statically balanced.
f. Find the new position of the center of mass. It started out at (0,0,0)
g. Find the linear momentum, P, of the system given the same rotation about the z axis.
Use the x-y-z coordinate system attached to the body and not cylindrical coordinates.
h. Find dP/dt. What is its magnitude and direction. Explain what dP/dt represents.
Survey concept question: If you rotate a rigid body about a point which is not the center of mass is an
external force required to keep the axis of rotation from moving? Yes, No, Not sure.
3. This problem illustrates two important concepts. One is that the rotation rate vector omega can
be expressed in the xyz coordinate system attached to the rotating body. The second is that
when you rotate about something other than a principal axis, the system is dynamically
unbalanced and produces torques about other axes.
a. Shown in figure A is a slender rod. Find or look up the [I/G] matrix for a slender rod.
Convince yourself on the basis of symmetry arguments that the axes shown are principal
axes.
b. For rotation about the z axis only find the angular momentum vector {H/G} from
[I/G]{omega}. It should have only one term. Compute dH/dt verifying that you get just
the expected Izz*omega-z dot term.
c. Now for the interesting part: Incline the rod at 45 degrees as shown. Spin it about a
vertical axis as shown at a rotation rate omega. Compute H/G=[I]omega. To do so you
will need to express omega in terms of vector components in the rotating frame
principal axes. Omega=omegaxihat+0jhat+omegazkhat. There is no rotation
component in the jhat direction.
d. Compute dH/dt for this situation. To do this you will need to use what you learned in
example 3.5 in Williams from problem 1. This results in multiple terms representing
torques in various directions. Which ones represent dynamic imbalance and which
account for the torque required to increase the rate of rotation.
Survey question: Which conditions are required to for a rotating body to be dynamically balanced:
rotate about a principal axis, rotate about an axis passing through the center of mass, both.
4. A solid sphere is shown, with mass M. It has an inextensible tape wrapped around it which is
connected to a spring. The spring has a spring constant K.
a. How many independent coordinates are required to completely describe the motion.
This is another way of asking how many independent degrees of freedom the problem
has.
b. Find an equation of motion for the system which accounts for rolling motion in the
plane of the paper. There is no slip. It is suggested that you choose a coordinate that is
zero at the unstretched position of the spring.
c. Find the natural frequency of the system.
Survey question: One approach to doing this problem is to compute the sum of the external torques
about the point of contact of the sphere with the ground. If you use this approach then you must also
compute the quantity dH/dt in order to find the equation of motion. With respect to what point should
one compute H. a. the center of mass of the sphere, b. the point of contact with the ground, c some
other point.