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cbasst
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Homework Statement
A cylinder of radius r, mass m, and rotational inertia 1/2mr2 slides without rolling along a flat, frictionless surface with speed v0. At time t = 0 the object enters a region with friction (with coefficients μk and μs), as shown above. Initially the cylinder slips relative to the surface, but eventually it begins to roll. Set t = 0 when the object enters the region with friction.
After the object enters the region with friction, but before it begins rolling without slipping, what is v(t), and what is ω(t)? When does the transition to rolling without slipping occur?
Homework Equations
F = ma
Ff = μN
[itex]\tau[/itex] = I[itex]\frac{d\omega}{dt}[/itex] = r x F
v = r[itex]\omega[/itex]
The Attempt at a Solution
To find v(t),
[itex]\Sigma F = ma[/itex]
[itex] a = \frac{-F_f}{m} = \frac{-u_kN}{m} = \frac{-μ_kmg}{m} = -μ_kg[/itex]
[itex] v = v_0 + \int a \ dt = v_0 - u_kgt[/itex]
To find ω(t),
[itex]\Sigma \tau = I \frac{dω}{dt} = -F_fr[/itex]
[itex]\frac{1}{2}mr^2\frac{dω}{dt} = -μ_kmgr[/itex]
[itex]\frac{dω}{dt} = \frac{-2μ_kg}{r}[/itex]
[itex]ω(t) = \frac{-2μ_kg}{r}t[/itex]
While my equation for v(t) agrees with the answer my book gives, the book has [tex]ω(t) = \frac{2μ_kg}{r}t[/tex] which omits the negative sign. I believe that this may be an incorrect omission on the book's part. It seems to me that if the force is happening in the [itex]-\hat{x}[/itex] direction and the radius is pointing in the [itex]-\hat{y}[/itex] direction, the torque should be in the [itex]-\hat{z}[/itex] direction. This implies that the angular velocity is increasing in the negative direction, so there should be a negative sign in the ω(t) function (how I did it). I would be happy to leave it there - case closed - except that it gives me a negative time for when the cylinder starts rolling without slipping (I set the v(t) and ω(t) functions equal, then then it gives me that [itex]t = \frac{v_o}{-μ_kg}[/itex]. What's the problem?