Bowling ball rolling - velocity, acceleration, time

[mybrohshi5]

Homework Statement

A bowler throws a bowling ball of radius 11.0 cm along a lane. The ball slides on the lane with an initial speed of 8.10 m/s and an initial angular speed of zero (i.e. the ball is not spinning at all when it first makes contact with the lane). The coefficient of kinetic friction between the ball and the lane is 0.19.
The kinetic frictional force f_k acting on the ball causes a linear acceleration of the ball while producing a torque that causes an angular acceleration of the ball. When the center-of-mass speed V_cm has decreased enough and the spin rate has increased enough, the ball stops sliding and begins to roll smoothly without slipping.

A) For what length of time does the ball slide?
B) Over what distance does the ball slide?
C) What is the linear speed of the ball when smooth rolling begins?

The Attempt at a Solution

I am having trouble coming up with the equations to solve for these.

I found the linear acceleration of the ball from an earlier part a = 1.86 m/s^2

I also found the angular acceleration of the ball from an earlier part [itex]\alpha[/itex] = 42.3 rad/s²

I know i need 3 equations and they will have three unknowns.

This is what i have came up with

V = a*t

[itex]\omega[/itex] = [itex]\alpha[/itex]*t

Are these two right for what i am trying to find? And any help on finding a third equation would be greatly appreciated :)

Thank you

 

[tiny-tim]

Hi mybrohshi5!

(have an omega: ω and an alpha: α and try using the X² tag just above the Reply box )

A) For what length of time does the ball slide?
B) Over what distance does the ball slide?
C) What is the linear speed of the ball when smooth rolling begins?[/B]

As you know, the speed of the centre of mass depends only on the (net) force, and the angular acceleration about the centre of mass depends only on the torque about the centre of mass …

now use the definition of rolling, that it happens (ie sliding stops) when v = aω.

 

[mybrohshi5]

I am a little confused about your explanation and where v=aω comes from? the only equations i have seen dealing with angular velocity and accel are

t = Iα
v = rω

and various alternations of those of course :)

 

[tiny-tim]

oops!

I am a little confused about your explanation and where v=aω comes from?

oops!

sorry, i meant v = rω.

 

[mybrohshi5]

Oh ok thank you. Were my first two equations the right ones to use for this type of problem?

v = at
ω = αt

t being time

Thank you for the help :)

 

[tiny-tim]

Yup! … you use F = ma with v = at,

and τ = iα with ω = αt.

 

[mybrohshi5]

Perfect Thank you very much tiny-tim