330M ASE Homework #1 (Review of Vectors & Dynamics)
Due: Friday, September 14th.
Reading Assignment: Read Chapters 1-3 of Torby (available as PDF from blackboard)
Problem #1:
During the encounter with the asteroid Gaspra, the Galileo spacecrafts position and inertial velocity
vectors, with respect to the origin of the solar system, were given by:
where O stands for the origin and S for the spacecraft. These vectors are measured in a Sun centered
inertial frame defined by unit vectors (i-j-k)
The position and inertial velocity vector of the center of Gaspra at that time were,
(a)
(b)
(c)
(d)
(e)
Calculate the position of the spacecraft with respect to Gaspra,
.
What is the magnitude of
Calculate the inertial velocity of the spacecraft with respect to Gaspra,
What is the magnitude of
Define unit vectors along
,
, and
. Assign names to each of
these unit vectors. Do these unit vectors form an orthogonal set? How do you know? Derive a
relation between the inertial vectors (i-j-k) and your rotating set of unit vectors.
(f) Express the position and velocity vectors
and
in terms of the rotating unit vectors in
(e). Is the velocity resulting
the as seen by an observer fixed in the rotating frame? Why or
Why not? Use the definition of the basic kinematic equation (BKE) to justify your response.
Problem #2:
In the figure below, a particle P is rotating with the disk. Assuming that
in both the frame and
and
find
frame components.
a) Write the position vector from O to P in terms of inertial coordinates and differentiate (without
using the BKE) to find velocity and acceleration,
and
b) Write the position vector from O to P in terms of rotating u-frame coordinates and use the BKE
to find the velocity and acceleration vectors,
and
.
c) Find the magnitude of the velocity in (a) and the velocity in (b). Do they match? Should they?
Problem #3:
Do sample problems 3.1 and 3.2(a) of Torbys text (available online). SHOW ALL WORK AND STEPS
REQUIRED IN ARRIVING AT THE FINAL SOLUTION. That is to say, even though the problems are worked
out for you, please show all intermediate steps necessary in arriving at the solution listed in each case.
Formulate the problem in terms of any set of coordinates you find most convenient (i.e. inertial or
rotating).