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mM G dp v F C r v C r ^ v F F ma dW F � dr acent ^ r r ^ dt r ;mM G U U mg h U k x^ K mv ^ K I r
mM G ;mM G X
Etot K U mv ^ ; L r � p I mi ri^ r a (^) i d m r m r v r T ^ ^ r r
q
k m Gm m dt
Z t
dL d r � F I T L I I F dt pf ; pi dt dt
q a
g l pr T ^ Ls GM Solid disk of mass M and radius R rotating ab out its cylindrical axis I ^ M R^
vf ; vi ;u ln mf ; g t I Icm M d^ Iz Ix Iy mi
v v f 0 f cos ^0 ; cos c c
Problem p oints
An unknown mass m hangs from a massless string and descends with an acceleration g The other end is attached to a mass m which slides on a frictionless horizontal table The string go es over a uniform cylinder of mass m and radius R see gure The cylinder rotates ab out a horizontal axis without friction and the string do es not slip on the cylinder Express your answers in parts b c and d in terms of g m and R
m 2
m 1
R
a = g/
a Draw free b o dy diagrams for the cylinder and the two masses b What is the tension in the horizontal section of the string c What is the tension in the vertical section of the string d What is the value of the unknown mass m
Problem p oints
A solid uniform disk of mass M and radius R is oscillating ab out an axis through P The axis is p erp endicular to the plane of the disk Friction at P is negligibly small and can b e ignored The distance from P to the center C of the disk is b see gure The gravitational acceleration is g
P
b
R
C
a When the displacement a n g l e is what then is the torque relative t o p o int P b What is the moment of inertia for rotation ab out the axis through P c The torque causes an angular acceleration ab out the axis through P Write down the equation of motion in terms of the angle and the angular acceleration As the disk oscillates the maximum displacement angle max is very small and the motion is a near p erfect simple harmonic oscillation d What is the p erio d of oscillation e As the disk oscillates is there any force that the axis at P exerts on the disk Explain your answer
