Three objects of equal mass, A, B, and C, are released from rest at the same instant from the same height on identical ramps

Three objects of equal mass, A, B, and C, are released from rest at the same instant from the same height on identical ramps. Objects A and B are both blocks, and they slide down their respective ramps without rotating. Object C rolls down the ramp without slipping. Its moment of inertia is unknown.
Objects A, B, and C are made of different materials, thus the coefficients of friction between the objects and their coffesponding ramps are not necessarily the same.

Object A reaches the bottom of its ramp first, followed by objects B and C, which reach the bottom at the same instant.
a. Rank the objects according to magnitude of center-of-mass acceleration, from largest to smallest. If any objects have the same magnitude center-of-mass acceleration, state so explicitly. Explain.
b. Rank the net forces exerted on the three objects according to magnitude, from largest to smallest. If the net force on arly two objects is the same, state so explicitly. Explain.
c. In the spaces provided, draw and label a (point) free- body diagram for each object.
Free-body diagraur for object i
F-ree-bocly dia*uram
for object B Free-body diagram
for object C
d. Rank the frictional forces exerted on the three objects according to magnitude, from largest to smallest. If the magnitude of the frictional force is the same on any two objects, state so explicitly. Explain your reasoning.

5. Energy analysis of falling-spools experiment
The modified Atwood’s machine shown at right consists of two identical spools connected by a massless, ‘ ” inextensible thread that runs over an ideal pulley. The thread is wrapped around spool A many times, but it is attached to a fixed point on spool B, so that spool B will not rotate.
The spools are released from rest from the same height at the same instant.
a. In tutorial, you observed the motion of the spools after they were released. Ignoring small dffirences in their nntions:
. In which direction did each spool move?
. Did spool A hit the ground before, after, or at the same instant as spool B?
Is the mngnitude of the center-of-nutss acceleration of spool A (while it is fallin g) greater than,less than, or equal to that of spool B? Explain.
Is the translational kinetic energy of spool A just before it hits the ground greater thsn, less than, or equal to that of spool B? Explain.
d. Is the total kinetic energy of spool A just before it hitsothe ground greater than, less than, or equal to that of spool B? Explain.
Consider the system consisting of all of these objects: spool A, spool B, the thread, the pulley, and the Eafth.
i. Explain how you can tell that the total energy of this system(i.e., Ugnv,o+ Ug^”,n * Kton,,e * K,*nr. n * Kro’ a, * K,o’ s) is constant as spools A and B fall.
b.
c.
Suppose that this system starts with Ugo”, o = f/srr, B = 9 J. Just before the spools hit the ground, which is where the zero for gravitational potential energy is chosen, spool A has translational kinetic energy K,on,”a = 4 J. Determine the value of the rotational kinetic energy of spool A at this instant. Show your work.
A third identical spool, spool C, is added to the falling-spools experiment described in the preceding problem.
As above, all spools are released from rest from the same height at the same instant. Spool C is not in contact with any other objects as it falls.
a, Rank the spools according to magnitude of center -of-mas s acc eleration (while falling), from largest to smallest. If any spools have the same center-of-mass acceleration, state so explicitly. Explain.
b. As in the preceding problem, suppose that Ugrrr. n = Ugr,, s = 9 J before the spools are released. Just before the spools hit the ground, which is where the zero for gravitational potential energy is chosen, spool A has trarrslational kinetic energy K,*,, e,= 4J.
i. Rank the spools according to maximum tanslatianal kinetic energy, from largest to smallest. If any spools have the same maximrrtn translational kinetic energy, state so explicitly. Explain. (Use the definition K*, = lma”^’.)
ii. Rank the spools according to maximum total kinetic energy, from largest to smallest. If any spools have the same maximum total kinetic energy, state so explicitly. Explain.

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