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Unformatted text preview: urnmi? _ ,. Physics 7A Midterm #2. A. Zettl, Fall 2006
All Problems worth 20 points. All four sides of two sheets notes (8—112" x 11") ok. Good luck. 1. A pipe of length L and mass M is partially ﬁlled with sand. It turns out that the sand is distributed
in such a way that its linear mass density inside the pipe is nonuniform and given by p(x) = Csin(xrtlL),
valid for O<x<L, with C a known positive constant. The central axis of the pipe is along the xaxis; one end
of the pipe is at x=0 and the other end is at x=L. Determine the x—coordinate of the center of mass of the
sand—ﬁlled pipe. SAND l S in; :51:
(EmN PIPE
—i\as MASS M)
2 Two spherical, uniform density "planets" in outer space have identical radii R and masses M. They interact via gravitational forces, but they do not move: Somehow they are each "nailed" to the x—axis
at speciﬁc locations. Planet #1 (of mass m1 = M) is always centered on the origin, while planet #2 (of mass
[112 = M) is always centered at xzd. A small rock of known mass m3, originally on the surface of planet #1
at x=R, is by external means moved along the x—axis from X=R to x=(d—R), i.e. directly across to the
surface of planet #2. a) Determine an expression for U(x), the total gravitational potential energy of the threemass system,
where x is the position of the rock as it is being moved. Your expression must be valid only for R<x<(d—
R). (You should adopt the standard convention and deﬁne U=0 for a pair of masses when they are inﬁnitely far away from each other.) Also, plot Utx) for R<x<gdRl. ' . b) Assume you want to throw the rock from the surface of planet #1 and have it land on planet #2. With what minimum releaSe speed do you need to throw the rock (at sz) for this to happen? Mo , mesa 3. You have been asked to check the design of a new "Ultimate Limits" roller coaster that uses no
seatbelts or restraint bars. Assume the roller coaster has just one small car that starts from rest at height h
and slides without friction always along a curved track. The track executes, from ground level, two circular
loops with respective radii R1 and R2. Your job is to insure that children do not at any time fall out of their
seats, or experience forces larger than "lOg's". Determine the largest R1 and smallest R2 (in terms of h and
any other parameters you feel are needed) that will insure that the roller coaster operates within the
speciﬁed limits. . .1 4. Consider the three masses m,=M mzzM/2, and m3=M all perfectly lined up along the x—axis. mI has an initial velocity vo’i‘while m2 and 1113 are initially at rest. m1 ﬁrst hits m2 and they undergo a fully
elastic collision. a) Determine the velocities (magnitude and direction) of m1 and m2 just after they collide. b) After its collision with mum2 hits m3. m3 is made of clay and always undergoes only fully
inelastic collisions. Determine the velocities of m2 and m3 after their collision. c) Determine (i.e. show) if, after its ﬁrst collision with n12, In1 will ever collide with m2 again. If there
will not be a second collision, how might you adjust the mass of m3 (and what would the smallest
adjustment be) to make a second collision with ml occur? lemma grwma WAL 9 D A
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5. A massless but "goofy" spring, one end of which is ﬁxed to a rigid wall, has equilibrium length Lo and a rather unusual restoring force: Frestore = A): — Bxs, where A and B are known positive constants. A
mass M is attached to the free end of the spring and you slowly pull on the mass, extending the spring. Just as the end of the spring has been stretched from x=0 to x=xm, the mass is released (essentially from
rest). Determine a) The work you did in stretching the spring.
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