LS50B Problem Set #7
Due Friday, March 25, 2016 at 5 PM
Problem 1: Warm up
Let’s start by limbering up your physics muscles. Answer the following eight multiple choice questions. For
each question, write a one sentence explanation and, where appropriate, draw a diagram to explain your
choice.
1. In the absence of air resistance, a ball of mass m is thrown upward, such that its final height will be
20 m. When the ball is half way up (at 10 m), what is the net force on the ball?
(a) 2mg
(b) mg
(c)
(d)
mg
2
mg
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2. Relative to the ground, an airplane gains speed when it encounters wind from behind, and loses speed
when it encounters wind head on. When it encounters wind at a right angle to the direction it is
pointing, its speed relative to the ground below:
(a) increases
(b) decreases
(c) is the same as if there were no wind
3. When an increase in speed doubles the momentum of a moving body, its kinetic energy:
(a) increases, but less than doubles.
(b) doubles.
(c) more than doubles.
(d) depends on factors not stated.
4. When a spinning system contracts in the absence of an external torque (e.g., a figure skater on frictionless ice pulls in their arms), what happens to the rotational speed and angular momentum of the
system?
(a) Rotational speed remains the same; angular momentum decreases.
(b) Rotational speed decreases; angular momentum decreases.
(c) Rotational speed increases; angular momentum increases.
(d) Rotational speed and angular momentum remain unchanged.
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(e) Rotational speed increases; angular momentum remains unchanged.
(f) Rotational speed decreases; angular momentum remains unchanged.
5. Consider a ball rolling in a horizontal circular path on the inside surface of a cone. The normal force
on the ball
(a) is mg
(b) is always greater than mg
(c) may be greater or less than mg
(d) is always less than mg
6. The fact that the Moon always shows its same face to Earth is evidence that the Moon rotates about
its axis once per:
(a) day
(b) month
(c) year
(d) None of these: the Moon does not rotate about an axis.
7. The Moon is most responsible for Earth’s tides. Which pulls more strongly on the Earth and its
oceans?
(a) the Moon
(b) the Sun
(c) Both about equally
8. An inflated balloon with a heavy rock tied to it submerges in water. As the balloon sinks deeper and
deeper, the buoyant force acting on it:
(a) increases
(b) decreases
(c) remains unchanged
(d) need more information.
Problem 2: The tallest tree in the world
The tallest known living tree is a coastal redwood (Sequoia sempervirens), with a height of 115 m. Researchers
have nicknamed this specific tree Hyperion.
1. What is the minimum amount of work that must be done on 1 g of water to get it from the roots of
this tree to the leaves at the very top?
2. The xylem vessels, which you can think of as narrow tubes, carry water through the trunk of the tree.
One hypothesis is that trees use capillary rise to wick water through the xylem from the roots to the
top of the tree. If this hypothesis is correct, what should the radius of the xylem vessels be in order to
get water to the top of Hyperion? The surface tension of water is about γ = 70 mN/m.
3. A typical radius for a xylem vessel is about 50 micrometers. Now that you know this, how tall could
a tree be and still supply its top leaves with water using capillary rise? What does this mean for the
redwood?
4. Another proposed mechanism for water transport in trees is that evaporation and transpiration from
the leaves imposes a negative pressure difference between the top and the bottom of the tree, pulling
water up from the roots. What pressure difference would be needed in order to simply maintain a
column of unmoving water the height of Hyperion?
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5. In a 2004 Nature paper, George Koch and his coauthors found that during the day, the xylem pressure
difference between the ground and the top of a tree as tall as Hyperion is about -1.8 MPa, while at
night, it is -1.3 MPa. If at its tip, Hyperion has a diameter of about 12 cm, and the xylem vessels
are packed densely into a single ring about 1 cm in from the outer edge of the trunk, the tree would
have about 1200 xylem vessels at the top. As before, the viscosity of water is η = 1 mPa s. Given
this assumption, how much water is evaporating from the topmost leaves during the day? At night?
Report your answer in µL/hour.
A"
B"
h"
R"
R"
Liquid"1"
Liquid"2"
a"
Liquid"1"
Liquid"2"
Figure 1: A visualization of problem 3. A: part 1; B: parts 2 and 3.
Problem 3: Bobbing proteins
Consider a spherical protein with radius R, near an interface between two liquids, liquid 1 and liquid 2.
(Perhaps it is at the surface of a cell). The surface tension between the protein and liquid 1 is γp1 , between
the protein and liquid 2 it is γp2 , and between the two liquids it is γ12 . The density of the two liquids are
equal, so you may disregard buoyancy for the purposes of this problem.
1. Write down an expression for the energy of this system if the area of interface between the two liquids
is A, and the protein is completely submerged in liquid 1, as in Figure 1A.
2. Write an expression for the energy of the system if the protein is at the interface, such that some
fraction h of its diameter sticks out into liquid 2, as in Figure 1B.
3. At what value of h will the total energy of the system be minimized (and thus, the system will be in
equilibrium)? How would increasing γp1 change h? How would increasing γp2 change h? Does this
agree with your intuition?
Problem 4: Push me, pull you!
One type of motion at low Reynolds number is often called “Push me, pull you” motion. In this problem, we will build a neuromechanical model to better understand this technique for swimming at low
Reynolds numbers. For inspiration, first take a look at this simulation: http://iopscience.iop.org/13672630/7/1/234/media/swimmer.swf. The red model, the winner of the race, is a model Push Me Pull You.
Then watch this real organism, related to Euglena sp., swim! https://www.youtube.com/watch?v=91yFpzzz4O8
Look familiar?
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Extend"rod"
2"
1"
Exchange"
cytoplasm"
1"
2"
1"
Exchange"
cytoplasm"
2"
1"
2"
Retract"rod"
Figure 2: One possible time irreversible cycle of motion for the Push Me Pull You.
What’s a Push Me Pull You?
We can model this as an organism that effectively consists of two spherical balloons, connected by a rigid,
extendable rod. (You can consider the rod to have effectively no diameter and therefore to contain no volume
and to experience no drag). The volume of cytoplasm inside the entire organism (both balloons) is constant.
However, the organism can pass cytoplasm between the two balloons, inflating one while deflating the other
at the same time. It can also extend and retract the rod, changing the distance between the two balloons.
These two degrees of freedom in its motion (inflation/deflation of the balloons, and extension/retraction of
the rod) allow the organism to move at low Reynolds number.
How can we model the motion of the Push Me Pull You?
We will call the left hand sphere 1, with radius R1 , and the right hand sphere 2, with radius R2 . For this
problem, we will make use of a result from Avron, Kenneth and Oaknin, 2005. In their paper modeling a
R1 −R2
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Push Me Pull You organism, they derive that over any small time step dt: dX = 2(R
d` + 4π`
2 dV1 ,
1 +R2 )
where dX is the net displacement of the organism over the time step, d` is the change in length of the rod
over the time step, and dV1 is the change in volume of the left hand sphere over the time step. (Remember:
the total cytoplasm volume is conserved, so dV1 = −dV2 .) Note that this expression breaks down the motion
into two distinct components: one due to drag on the two spheres as the length of the rod changes, and one
due to pressure in the fluid as the spheres expand and contract.
Building your neuromechanical model
1. First, we will consider how the organism will “know” when to take steps. Note that while we talked
about central pattern generators as circuits of neurons in class, for a single celled organism the central
pattern generator could instead be a molecular oscillation that controls the behavior. Here, we will use
a central pattern generator with two independent controls: one for inflation/deflation and one for rod
extension/retraction. For simplicity, we will assume that the organism is always inflating or deflating
with some constant rate, and the rod is lengthening or shortening at another constant rate - so we
only need to worry about whether we are inflating or deflating, and whether the rod is lengthening
or shortening. Since each control has only two possible states, we may consider them to simply be
0 at some times and 1 at others. Describe at least one way you could create a control function that
exhibits this behavior of switching between high and low states (no need to use the Heaviside Function
discussed in class).
2. Now, consider how you would simulate swimming by updating the organism’s changing rod length and
balloon volume as you take small steps in time (dt). How you would combine the equation above with
your central pattern generator to simulate the animal taking steps? Which values in the equation will
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need to be updated at each time step? Using comments in MATLAB, create an outline for how you
would calculate the change in rod length and balloon volume at each step in time. Think about how
this will depend on the states of your CPG. You may assume that ` and V increase or decrease with
constant rates while under control of the CPG and that they are capped by minimum and maximum
allowed values due to morphological constraints on the organism’s body.
3. Time to implement your algorithm! To keep things simple, let’s assume that 4µm< ` < 12µm for the
entire rod and that 4µm3 < V < 16µm3 for any single balloon. For the magnitude of rates, let the rate
of rod extension/retraction be 0.02 µm/ms and the rate of volume change be 0.02 µm3 /ms. Simulate
the PMPY swimming over 20 s taking 1 ms steps. Plot the position of your animal as a function of
time. Does it move? Does it go in the direction you would expect? Does it move in a single direction
throughout a cycle of your CPG, or does it have more of a “two steps forward, one step backward”
approach?
4. Now let’s try simulating the swim with a few different values of phase. Run your model where the
phase between the two CPGs is shifted by − π2 , 0 and π2 . Under what conditions do you get motion?
Can you explain why this might be?
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