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Unformatted text preview: 3/11 Mean260 2  E 2
Std. Dev: 20 Phil “am Name: , Slot: 0 You must clearly state which ’truths’ (a particular conservation law, Newton’s 2nd or 3rd
Law, etc.) you are using each time you use them. Also, for truths that are only SOMETIMES
true (like conservation laws), you must justify the use of that law in the particular problem. 0 Pay attention to point values — use your time wisely. 1. [15 points] A block starts at the right end of a track as shown. The block first slides down a curved section
of frictionless track, then across a frictionless horizontal section, and finally up a straight inclined plane section
that is NOT frictionless. The coefficient of kinetic friction between the block and the incline is pk.
a) Use energy methods only (no; Newton’s 2”d Law + kinematics) to show that the maximum height reached
by the block on the left side is ymax = ——h——. Yes, do the algebra. [10 points] 1+ ,uk cot(l9)
b) Could the behavior of this system during the entire trip from right to left end he explained with the aid of
a potential energy diagram? Explain. [3 points]
c) Having come to rest at ymax, will the block start to slide back down the incline? Explain. [2 points] This left section l_§ perfectly straight. It is shown with a gentle curve at the beginning
to make it clear that the block doesn’t 'crash’ into the inclined part of the track. 2. [15 points] Shown below is a half sphere of radius R. The sphere is solid (not a thin shell) and has uniform
density p. Prove that the coordinates of the c.o.m. are (BR/8, O, 0). Setup and solve the integral required to
get this answer. l1 / I *7 ‘
3. [15 points] A cannon (shown below) is rigidly attached to a carriage, which can only move forward (right)
and backward (left) and is connected to a large spring, with spring constant k, that is initially unstretched,.
The mass of cannon + carriage is M. The mass of the projectile fired by the cannon is m. The projectile is fired at an angle 9 up from the horizontal, at velocity v. a) Assuming that there is no friction between the carriage wheels and the ground, find the maximum
extension of the spring. Carefully (bUt briefly) justify each step of your work. [10 points] b) Of course the cannon will recoil after firing a projectile. Could you design a gun that has no recoil (or has
very little recoil)? Be creative. No ”cheating" — the gun is expected to fire projectiles of large mass at high
velocity! It.shouldn’t take more than a sentence or two, and perhaps a sketch, to specify your invention. [5 points] ' 4. [8 points] Two barges (shown below) are moving in the same direction in 'still’ water (i.e., no current) at
different (but constant) speeds. The faster barge (the upper one) is about to pass the slower one. Coal is
tossed from the slower barge to the faster one at a constant rate (some number of kg of coal per minute). It the barges are not to change sgeed as the coal is transferred... . a) Does the engine on the faster barge have to provide additional force (compared to before the transfer)?
b) Does the engine on the slower barge have to provide additional force (compared to before the transfer)?
Assume that the shoveling is perfectly sideways and frictional forces1 between the barges and the water do
not depend on the mass of the barges. You can 'talk’ your way through this problem, but you may find it easier to do a simple (two line!) calculation 
assigning numbers or variable names to masses and velocities of the barges and the amount of coal tossed in
one minute. A simple sketch (representing the boats and the coal as point masses, with specific V’s) will very
likely help. ' 1 Clearly there is water friction — before the transfer the engines push the barges through the water, yet the barges aren’t
accelerating.
V 5. [42 points] Shown below is a massless ideal spring of spring constant k and equilibrium length 2H. The spring was compressed so it could fit between a ceiling and floor that are a distance H apart. One end of the spring is permanently fixed to the ceiling and the other end is permanently connected to a
small block of mass m and negligible height. The block can slide to the right or left on the frictionless floor. (This is a onedimensional problem so the block can only move right or left.) The figure shoWs the configuration at t = O; the block is moving to the right at velocity v and the block’s center
is not directly below the spring’s connection point to the ceiling, but is displaced to the left a distance 8 that is small compared to H. The displacement of the block from center initially takes the value 8, but more
generally (at later times) the displacement may be denoted as ’x’. Assume the block never loses contact with
the floor. (This is true for small t, but not necessarily for large t. You should assume it is true at all times.) a) Is mechanical energy conserved? Why/why not? [3 points] b) Is momentum conserved? Why/why not? [3 points] c) What is the potential energy of the system, U(x), as a function of the parameters given above (k, H, etc.)? [8 points]
cl) Sketch U(x) for positive and negative x. You can use calculus, but it may be easier to use common sense. [8 points]
e) Find the force felt by the mass, F(x), directly from U(x). Do the math. (No credit for other approaches.) [5 points] f) If at t = 0, displacement = e and velocity = +v, will the block ever be found to the right of center? If you cannot give a
'yes’ or 'no’ answer, provide a relationship between system parameters that will answer the question once numbers aig_,
plugged in. [5 points] g) For this part of the problem, assume that at t = 0, displacement = e and the block is at rest. Will the block ever be
found to the right of center? Explain. [3 points] ‘ h) Describe the motion of the system over time (as we did in class for other systems) for the case where the initial
velocity v is large (but finite!) and then for the case where v is very small (but not zero). Include a brief discussion, but
M calculations, of equilibrium points (if any), turning points (if any), etc. [7 points] i) Extra credit — Assume the block starts with displacement+e and velocity +v and ”just makes it” to displacement = 0.
How much work is done by the force F(x) you found in part e during this motion? For credit you must do this two ways —
with an integral (setup the integral, but don’t solve it) and without (setup, but don’t solve the algebra). 6. [5 points] Feynman talks about taking ‘nature walks’ with his father. There is a discussion about the name of a
particular breed of bird they encounter. His father says... ”Do you know what that bird is? lt’s called a brown—throated
thrush. But in Portuguese it’s called a ...; in Italian it’s called a ; and in Chinese it’s called a...” What is the point of the
story? (It’s a point I’ve tried to make several times in this course.) If you don’t remember the bird story, consider the ”ball in the wagon” story. Feynman carefully observes how a ball
rolls in a wagon when the wagon starts and stops. His father’s response to ”why does the ball behave this way?” begins
with ”it’s called inertia, ...” What is the point of the story? (It’s pretty much the same point as the bird story.) A single sentence will answer this question... ...
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 Spring '08
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