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**Hi everyone, this problem involves smooth rolling and translational motion:**

1. Homework Statement

1. Homework Statement

A bowler throws a bowling ball of radius R= 11 cm along a lane. The ball slides on the lane with initial speed v

_{com}= 8.5 m/s and initial angular speed w

_{0}= 0. The coefficient of kinetic friction between the ball and the lane is 0.21. The kinetic frictional force acting on the ball causes a linear acceleration of the ball while producing a torque that causes and angular acceleration of the ball. When speed v

_{com}has decreased enough and angular speed w has increased enough, the ball stops sliding and then rolls smoothly. d)how long does the ball slide?

## Homework Equations

F

_{linear}= ma = -mgμ

_{k}

torque = rF = Iα

K

_{i}+ U

_{i}= K

_{f}+ U

_{f}

## The Attempt at a Solution

I have figured out the linear acceleration a = -2.1 m/s

^{2}, and angular acceleration α = 47 rad/s

^{2}using the fact that F

_{linear}= ma = -mgμ

_{k}and torque = rF = Iα.

I know that there is smooth rolling if v

_{com}= rω = (.11m)ω

I set up an equation using conservation of energy to solve for ω

_{f}, which is whenvsliding should end and smooth rolling should begin.

(1/2)mv

_{i}

^{2}+ (1/2)I ω

_{i}

^{2}= (1/2)mv

_{f}

^{2}+ (1/2) I ω

_{f}

^{2}

(1/2)m(8.5m/s)

^{2}= (1/2)m(.11ω)

^{2}+ (1/2)(2/5 mr

^{2})ω

_{f}

^{2}

Canceling out mass and simplifying:

36.125 m

^{2}/s

^{2}= .00605ω

^{2}+ .00242ω

^{2}

ω

_{f}= 65.3 rad/s

then, ω = ω

_{0}+ αt

65.3 rad/s = 47 rad/s

^{2}t

t= 1.4 s

However, the given answer in the book is 1.2s, and I'm not sure what's going on...I've tried to account for rounding errors, but that doesn't appear to be the problem. Am I neglecting friction when I shouldn't be, and if so, how would I calculate energy lost to friction if I don't yet know the distance the ball traveled? Any help would be appreciated, thank you.