d. Now assume that the bumper has a nonnegligible mass. After the col compressed

Question

d. Now assume that the bumper has a nonnegligible mass. After the col compressed to a maximum distance of about 90% ofthe value ofx in for this decrease sion with the bumper, the spring is part c.. Give a reasonable explanation M. R 2006M3 A thin hoop of mass M, radius R, and rotational inertia MR is released from rest from the top of the ramp of length Labove. The ramp makes an angle with respect to a horizontal tabletop to which the ramp is fixed The table is a height Habove the floor. Assume that the hoop rolls without slipping down the ramp and across he table. Express all algebraic answers in terms of given quantities and fundamental constants. Derive an expression for the acceleration of the center of mass of the hoop as rolls down the ramp. b. Derive an expression for the speed of the center of mass of the hoop when it reaches the bottom of the ramp c. Derive an expression for the horizontal distance from the edge of the table to where the hoop lands on the floor. Suppose that the hoop is now replaced by a disk having the same mass M and radius R. How will the distance from the edge of the table to where the disk lands on the floor compare with the distance determined in part c for the hoop Less than The same as Greater than Briefly justify your response.

Explanation / Answer

A) a = alphaR = Torque about point of contact / i

= MgR^2 sin theta / 2MR^2 = g sin theta/2

B) by third equation of motion,

v = sqrt (2as)

= sqrt(g L sin theta)

C) by second equation of motion, time of flight = sqrt(2H/g)

Horizontal distance = vt = sqrt(2H/g*gL sin theta)

= sqrt(2HL sin theta)

D) distance would be greater than as translational kinetic energy would be greater due to less rotational KE due to less moment of inertia

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