Peoples Physics Concepts

http://www.ck12.org Chapter 9. Rotational Motion

Στ=Iα start Newton’s second law for rotation

f r=Iα substitute in friction for the net torque

f r=I

∆ω

∆t

substitute in the definition of angular acceleration.∆tis the time it takes for the ball to begin rolling

f r=I

ωf−wo

∆t

substitute inωfandωofor∆ω

f r=I

ωf

∆t

ωois zero so we an simplify the equation

f r=I

vf

r∆t

sinceωfis the angular speed when the ball begins rolling without sliding, we can express it in terms ofvf

vf=

f r^2 ∆t

I

solving forvf

Now, we can put that value we just found back into our Newton’s second law equation and solve for∆t.

f=m

f r^2 ∆t

I −vo

∆t

making the substitution

∆t=

mvo

m f r^2

I −f

solving for∆t

∆t=

mvo

μkm^2 gr^2

I −μkmg

substitute the known values forf

∆t=

mvo

μkm^2 gr^2

(^25) mr 2 −μkmg
substitute the known value forI
∆t=

2

3

vo

μkg

simplify to get the answer

Watch this Explanation

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Explore More

1. A solid cylinder of mass, M, and radius R rolls without slipping down an inclined plane that makes an angleθ.
   The cylinder starts from rest at a height H. The inclined plane makes an angleθwith the horizontal. Express
   all solutions in terms of M,θ, R, H, and g.
   a. Draw the free body diagram for the cylinder.