A disk is mounted with its axis vertically. It has radius R and mass M$.$ It is initially at rest. A bullet of mass m

and velocity v is fired horizontally and tangential to the disk. It lodges in the perimeter of the disk. What

angular velocity will the disk acquire? $(2$

$(+2))$

$4.$ A small mass m slides from rest down a smooth slope, dropping a distance in elevation. It strikes the end of

a rod length d and mass M and sticks to it$.$ The rod is initially at rest and fastened by a pivot P$.$ How fast is

the small mass m moving just after the collision? $(32$

$3+)$

$5.$ The evolution of a star depends on its size. If a star is sufficiently large, the gravity forces holding it together

may be large enough to collapse it into a very dense object composed mostly of neutrons. The density of

such a neutron star is about $1014$ times that of the Earth. Suppose that a star initially had a radius of $7$ x $108$

km$,$ and it rotated once every $26$ days. What would be the period of rotation if the star collapsed to a radius

of $15$ km$?(1\mathrm{.}19$ x $10-14$ days$)$

$6.$ A hoop of radius RH and mass mH and a solid cylinder of radius RC and mass mC are released simultaneously

at the top of a plane ramp of length L inclined at angle above the horizontal plane. Which reaches the

bottom of the plane first, and what is the speed of each? $($solid cylinder$)$

$7.$ An athlete whirls a discuss in a circle of radius $80\mathrm{.}0$ cm$.$ At a certain instant, the athlete is rotating at $10$

rad$/$s and the angular speed is increasing at $50\mathrm{.}0$ rad$/$s$2.$ For this instant, find the tangential and centripetal

components of the acceleration of the discuss and the magnitude of the acceleration. $(89\mathrm{.}4$ m$/$s$2)$

$8.$ You are designing an airplane propeller that is to turn at $2400$ rpm$.$ The forward airspeed of the plane is to

be $75\mathrm{.}0$ m$/$s$,$ and the speed of the propeller tips through the air must not exceed $270$ m$/$s $($This is about

$80\%$ of the speed of sound in the air. If the propeller tips moved faster, they would produce a lot of noise$)$

a$.$ What is the maximum possible propeller radius? $(1\mathrm{.}03$m$)$

b$.$ With this radius, what is the acceleration of the propeller tip? $(6\mathrm{.}5$x$104$ m$/$s$2)$

$9.$ We wrap a light, nonstretching cable around a solid cylinder, of mass $50$ kg and diameter $0\mathrm{.}120$ m$,$ that

rotates in frictionless bearings about a stationary horizontal axis. We pull the free end of the cable with a

constant $9\mathrm{.}0$ N force for a distance of $2\mathrm{.}0$ m; it turns the cylinder as it unwinds without slipping. The

cylinder is initially at rest. Find its final angular speed and the final speed of the cable. $(1\mathrm{.}2$ m$/$s$)$