Chapter 11, Problem 016 Nonuniform cylindrical object. In the figure, a cylindri

Question

Chapter 11, Problem 016 Nonuniform cylindrical object. In the figure, a cylindrical object of mass M and radius R rolls smoothly from rest down a ramp and onto a horizontal section. From there it rolls off the ramp and onto the floor, landing a horizontal distance d 0.475 m from the end of the ramp. The initial height of the object is H = 0.95 m; the end of the ramp is at height h = 0.12 m. The object consists of an outer cylindrical shell (of a certain uniform density) that is glued to a central cylinder (of a different uniform density). The rotational inertia of the object can be expressed in the general form 1-BMR2, but is not 0.5 as it is for a cylinder of uniform density. Determine .

Explanation / Answer

it leaves ramp horizontal,

so vertical velocity at that time, v0y = 0

y = 0 - h =- 0.12 m

ay = - 9.81 m/s^2

y = v0y t + ay t^2 / 2

- 0.12 = 0 - 9.81 t^2 / 2 => t = 0.156 s

d = v t

v= 3.04 m/s

Now applying energy conservation ,

PEi + KEi = PEf +KEf

m g H + 0 = m g h + ( m v^2 / 2 + I w^2 / 2 )

m g (H - h) = m v^2 /2 + (beta m r^2) (v/r)^2 / 2

2 g (H - h ) = v^2 + beta v^2

2 x 9.81 x (0.95- 0.12) = 3.04^2 (1 + beta)

beta = 1.77 - 1 = 0. 77 ......Ans

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