1) A particle of mass m and initial speedv0 collideswith and sticks to the edge

Question

1)A particle of mass m and initial speedv0 collideswith and sticks to the edge of a

uniform solid disk of massM and radius R. If the disk is initially at rest and is

pivoted about africtionless axle through its center (perpendicular to thepage):

(a) find the angularvelocity of the system after the collision.

2)A uniform solid disk of radius R and massM is free to rotate on a frictionless

pivot through a point onits rim. If it starts from rest with its center at thesame

height as thepivot:

(a) What is its angularacceleration both at the start and when itscenter is directly

below the pivot?

(b) What is its angularvelocity both at the start and when itscenter is directly below

the pivot ?

(c) What are thecomponents of the reaction force of the pivot on the disk both atthe

3)Suppose that we hang from its rim a hoopof diameter 30cm and mass .4kg and set it swinging.

a)Define the moment of inertia formally andfind it for the given hoop. b) State thefundamental

dynamical equation ofrotational motion. c) Now applythe equation to this problem carefully

inserting all thespecifics as far as you know them. d) In thelimit of small displacements solve the

Assuming that the spoolrolls withoutslipping:

(a) What is theacceleration of the center of mass ?

(b) What is its center ofmass velocity after rolling a distance d ?

5A ballistic pendulum consists of a simpleuniform rod of mass M and length L hanging vertically at rest.A

blob of putty of mass m isshot horizontally at speed v so that it collides with and sticks tothe lower hanging

end of the rod at itslowest point. Find the maximum angle that the rod now swingsthrough as it swings

upward recoiling.

Explanation / Answer

I can help with some of these. For #5: m1v1 = (m1+m2)v2 1/2mv^2 = 1/2(m1+m2)v2^2 + (m1+mg)gL(1-cos) Solve for cos. For #4 T = r X F Sum all forces F = ma F + ff = ma Sum all torques R(F - ff) = I(a/r) plug in the inertia, divide by R F- ff = (1/2)ma Add the forces and this torque equation to remove the frictionalforce, ff 2F = (3/2)ma a = 4F/3M This is the center of mass acceleration To find the velocity about the center of mass, use kinematics vf^2 = 2*a*d, solve for vf Use this a result to solve for ff F - f = ma F - f =m(4F/3m) F-f = 4F/3 f = F/3 might be negative, btw. #3. Sigh... okay. When you hang a hoop from its rim, you need to changethe inertia to its axis of rotation I = Icm + mR^2 where R = R+thickness of the hoop, call it "d" so R = R+d look in your book for inertia of a hoop about its center of mass...i dont remember it :( this is about what i remember for this problem.. you'll need tocheck your book for whatever this "fundamental equation ofrotational motion" is.. it's probably T = IA for small displacements, it means sin = and cosneeds to be expanded as a series using binomial theorem man im feeling tired. i apologize for not being more helpful...

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