Physics 116a02: Extra Hints for Assignment 10
Department of Physics and Astronomy, Vanderbilt University
This hints page was completed on March 7.
Things to Think About for Assignment 10
- Angular Motion with Constant Acceleration
- The angular acceleration variable alpha is assumed to be constant
but not zero.
- For part E: In my notes I always write the explicit parameter dependence
in the variable on the left size of the equals sign: x(t), for example
in one dimensional motion.
- For part F: All three of the option answers may be true for a specific
case of rotation, but only one answer is always true for any kind of
rotation (including rotation when the angular acceleration is not constant).
- For part G: You should assume that t1 can be different from 0.
After you give the correct answer, the program will say "Also, it [particle B]
has been in motion for less time than particle A." That statement is true
only if t1 is a positive, non-zero number.
- Scaling of Moments of Inertia
- In the "Learning Goal" description, the explanation of the dimensionless
constant c is not too enlightening. The constant c is a pure, positive
number whose magnitude depends on the shape of the rigid body and the placement
of the axis of rotation. The same rigid body can have two different values of
the moment of inertia if there are two different positions for the axis
of rotation. Even worse, the last two sentences in the "Learning Goal"
description should be qualified to say that the positions of the axis of
rotation are the same when the comparisons are being made. One can never
make a statement about the value of the moment of inertia unless the
position of the axis of rotation is specified.
- For part A: The axis of rotation is specified in this question as going
through the center of the shell: the center-of-mass is the center for a
spherical object of uniform density. The orientation of the axis is not
specified, but that does not matter for a spherical object. All orientations
are equivalent because of the symmetry of the spherical object. Lastly,
the object is a spherical shell, but your answer for the mass would be the same
if two solid spheres were being compared instead.
- For part B: Unless you can do the required calculus in your head, you
should look up two equations for the moment of inertia of a disk and of a
sphere. What is the exact ratio for Idisk/Isphere?
- Introduction to Moments of Inertia
- For part A: Be aware that you may have to click on several of the answer
choices, not just one. Also, "total mass" and "shape and density of the object"
is slightly redundant. If you know the shape of the object, and the density
of the object, then you can calculate the total mass in principle.
- For part B: Not a trick question, but one which addresses the
defects which I mentioned in the "Learning Goal" description of
the previous problem.
- For part C: The values of the masses for a and b were
given in the part B question. Be careful when you look at the figure for
the distances of a. At first it might look as if a is labeled 3r
units above the x axis, but it is only r units above the
x axis. Similarly a is 3r units away from the
y axis, not r units away.
- For part E: Don't confuse the lower case Latin letter w with the
lower case Greek letter pronounced as omega.
- Circular Motion Tutorial
- For part A: Since the t dependence is explicit on the left side
of the equals sign, then you should make t appear in the right side
of the equals sign.
- For part B: The initial value of theta at t=0 is assumed to
be 0.
- For part E: Before submitting your answer, check that is correct at
t=0 in that it gives a vector pointing along the y axis,
and that is correct for times just slightly greater than 0 at which time the r
vector should have a negative value for its x component (assuming that
omega is a positive number).
- Constrained Rotation and Translation
- In the first figure, the layers of tape are the thin black lines, and the
tape layers are wound around the orange colored drum. The tape layers
are all assumed to be very thin, much thinner than the diagram is showing.
So all the tape layers can be assumed to be at the same radius value r.
- For part B: As the problem statement indicates, you do have to worry
about sign conventions. In problem B, the value of omega(t) is
stated to be negative. So with this negative sign already included, does your proposed
answer indicate that the tape is moving to the left (positive result)
or to the right (negative result) as the tape is being wound back onto
the drum?
- For part D: Change to the new figure. The small, incomplete circle with
the red + sign inside does not necessarily indicate
the direction that the tire is rotating. Instead this circle indicates the
direction of positive rotation. You should also realize that your answer for
omega should be in units of inverse seconds, or more exactly radians/second, since
radians does not have any dimensions.
- For part E: Again think that if the acceleration is to be a positive
value (going faster to the right), which way is the tire rotating? Is that
direction of rotation a positive direction or a negative direction?
- An Exhausted Bicyclist
- For part A: First, before attempting to solve this problem,
answer the following questions to make sure that you understand
the equation given in the problem description.
In this expression for the angular velocity of the
bicycle wheel, what are the units of the 2 coefficient in sin(2t)?
Similarly, what are the units of
two fractions 1/2 and 1/4? Remember that the omega is in units
of radians/second. How do you obtain theta(t) if you are
give omega(t)? Be careful of the initial condition at $t=0$.
- For part C: The answer is in units of cm.
- Visualizing Rotation
- Make sure that you run the two applets in the problem description on the
left. For the first applet, at 0 angular acceleration, it is useful to
compare the top and the middle motions, before looking at the bottom motion
which is in the opposite direction. Remember to hit the reset button
before giving another hit of the run button.
- For part A: You will certainly have to run the applet more than once,
unless you have an extraordinary photographic memory. Also, this is a slightly
tricky question (which it should not have been). The question asks for which
rotations have positive initial angular velocity, not which have
positive angular velocity for most of the rotation time. So you have to look
carefully to see if any of the rotations start out with zero initial angular velocity,
which is not so easy to spot. And zero is not a positive, nor a negative,
number. Make sure that your submitted answers do not contain a comma or a space
between the numbers.
- For part B: Same caution applies as in part A.
- For parts C to E: What would be the signature of a rotation which had
zero angular acceleration over the whole rotation time? Similarly what would
be the signature of a rotation for which there was a constant
positive angular acceleration? What about a constant, negative angular acceleration?
- Graphs of Linear and Rotational Quantities Conceptual Question
- Essentially this problem shows 8 graphs, all having
the time parameter as the independent (horizontal axis)
variable. Your job is to figure out what is the dependent
variable (vertical) axis, given the description of the rotating
yellow dot on the wheel. It might be wise for you to review what
the answers were in the Circular Motion tutorial (item 4) before answering
the questions in this tutorial.
- After you submit your answers, but before doing the submit item,
you should do a mental exercise to see how your answers would change
if the wheel was rotating in a clockwise direction instead of a
counter-clockwise direction.
- Kinetic Energy and Moment of Inertia
- For part C: Do you understand this answer in terms of the answer
to part B?
- Ladybugs on a Rotating Disk
- In the figure, the x axis and the y axis in the
horizontal plane are
fixed in space. They are not drawn on the rotating disk.
Which type of friction (static or kinetic), if any, is operating between the feet of
the ladybugs and the surface of the disk? Is there any horizontal force
acting on the ladybugs? If so, what is the direction of that force and
what is causing that force?
- For part C: We discussed the direction of the angular velocity vector
in class, according to the right hand rule.
- Loop the Loop with a Twist
- This problem should explain why in the class demonstration of
the loop-the-loop apparatus, we had to release the rolling ball
at a significantly different height from the expected 5R/2,
or did we? I'll show the demonstration again on Thursday.
- For part A: For the last answer option "depends on the moment of
inertia", you should interpret that as meaning that any of the three other
answers could be correct depending on the exact value of the moment
of inertia of the rolling ball.
- For part B: This is a non-trivial calculation, and 11 (!) hints are provided.
- Pushing a Merry-Go-Round
- This problem is the inverse of item 6 for which you were given the angular
velocity as a function of time and then asked about the angular displacement.
Here you are being given the angular displacement function of time.
- For parts B to D: The answers should be a number, not a Greek symbol.
- Two Faces of Velocity
- For part A: The answer is requested to three significant digits,
but the program complains if only three significant digits are given,
and then shows a four significant figure answer instead.
You can give 4 significant digits.
- For part B: A complicated description for what should be a very
easy answer for you by now. The answer being requested is
just the magnitude of the linear velocity. You should realize
that the direction of the linear velocity vector is not the same
as the direction of the angular velocity vector.
- For part C: Three significant digits works for this part, as requested.
- Rotational Kinematics Ranking Task
- For part A: The question does not specify any particular time
(initial time, final time, or somewhere in between) when you are asked
to rank the angular velocities. What does that not make a difference
for these particular graphs?
- For part B: Think about the answer which you gave to the question
which I asked for part A.
- Constant Angular Acceleration in the Kitchen
- Dario is really an Italian first name, as Dario Fo the winner of the 1997
Nobel Prize in literature, or Dario Argento who is an Italian screenplay
writer, director, and actor with several awards to his credit. The things
you can learn from Google.
- For part A: Give your answer to three significant figures, in degrees/second-squared.
- Rotational Kinetic Energy and Conservation of Energy Ranking Task
- This problem is the inverse of the class demonstration showing
the "race" down an inclined plane between a hoop and a solid disk
of the nearly the same mass and radius.
- Exercise 9.8, Rotating wheel
- We did not write omega_z in the class notes. The
z subscript just means that the direction of the angular
velocity vector is along the z axis, which is what
it will be if the rotation motion is only in the xy plane. Many
textbooks don't bother with the z subscript when
first introducing rotation in the xy plane.
- For part A: You could get confused for this answer since normally
we write v = r*omega. If omega is constantly increasing
(becoming less negative, being 0 instantaneously, and then increasing
with time), then you might think that v will also be constantly
increasing. However, the problem wants you to think about speed
as the magnitude of the linear velocity, and magnitudes are never
negative. So during the time period
when the v is going less negative to 0, the speed is actually
decreasing. When the v is increasing from 0 to more positive
numbers then the speed is increasing. Also the phrase "time interval"
is not a good description since interval normally means give the beginning
and the end time. However, this problem is asking for only one number,
which you can assume means the length of the time interval.
- Exercise 9.22: Compact Disc
- After working this problem, you will know somewhat of how a CD works,
except for the laser optics. CDs were once a leading edge technology
but now are becoming obsolete in many applications.
More modern music players use "Flash Memory",
which is sometimes mis-named "Flash Drive" implying incorrectly that it works like
a spinning disk. "Memory sticks" which work on USB ports of
computers are also examples of "Flash Memory". All are examples of stuff
made possible by solid-state physics advances in the last decade.
- Exercise 9.28
- Exercise 9.34
- Exercise 9.36
- Exercise 9.46
- Exercise 9.50
- Exercise 9.58
- You can use a two dimensional integration formula, analogous
to equation 9.21, which follows from the general equation 9.20.
The key is to set of pairs of mass elements dm in say parallel strips
which are the same distance x or y away from
the axis of rotation. Then you integrate over the distance variable
between limits of 0 and a/2 or b/2.
- Exercise 9.62
- The key again is to set up the integrand in terms of the distance
away from the rotation axis, assuming a uniform (linear) mass
density. That is, dm = lambda*dr where r is the
distance along the rod from the rotation axis, and the constant linear mass density
lambda = M/L, the total mass divided by the length of the rod.
- Exercise 9.74
- You can think of the lead foil as being a thin spherical shell.
Once you do this problem, can you think of an easy, non-calculus way to calculate
the moment of inertia of a sphere of radius R2 which is hollow
up to an inner radius R1, where R1 < R2 ?
- Test Your Understanding 9.2: Rotation with Constant Angular Acceleration
- Test Your Understanding 9.1: Angular Velocity and Angular Acceleration
- There must be two vertical different scales, in different units (rad/sec and
rad/sec-squared), for the green and the yellow curves. Ask yourself the
following: if you had only the green (omega) curve, would you be able to derive
the yellow (alpha) exactly? Similarly, if had only the yellow curve, would you
be able to derive the green curve, as shown? As for the z subscripts,
consult what was hinted about item 17.
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This page was last updated on March 7, 2008