Impulses connecting linear and rotational motion

Problem:

A perfectly uniform billiard ball at rest is struck by a cue at a height h.  This causes it to have an initial velocity v0 and an initial angular velocity ω0.  How far will it go on a surface with coefficient of friction μ before pure rolling sets in?

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Solution:

Problem:

A homogeneous thin rod of mass m and length 2a slides on a smooth, horizontal table, one end being constrained to slide without friction in a fixed straight line.  It is initially at rest, with its extension normal to the line, when it is struck at the free end with an impulse Q parallel to the line.
(a)  Determine the initial motion of the rod.
(b)  Show that the force exerted by the line on the rod is given by 
Q2sinθ/[ma((4/3) - sin2θ)2], where θ is the angle between the rod and the line.

Solution:

Problem:

A sphere of radius R, mass M, and radius of gyration k = (I/M)½ about any axis through its center rolls with linear velocity v on a horizontal plane.  The direction of motion is perpendicular to a vertical step of height h, where h < R.  The sphere and step are perfectly rough and inelastic.  Show that the sphere will surmount the step upon collision if v2 > 2ghR2(R2 + k2)/(R2 - hR + k2)2.

Solution:

Problem:

A man wishes to break a long rod of length l by hitting it on a rock.  The end of the rod which is in his hand rotates without displacement as shown in the figure below.

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The man wishes to avoid having a large force act on his hand at the time the rod hits the rock.  Which point on the rod should hit the rock?  (Ignore gravity).

Solution:

Problem:

A uniform disk of radius R and mass M is spinning about its diameter with angular velocity ω, as shown below.  Located on the rim of the disk, at an angle θ from the spin axis, is point P, and P is moving with speed vp.  Point P is now suddenly fixed.  Show that the subsequent linear speed vc of the center of the disk is vc = vp/5.

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Solution: