What is the time tR for a cylinder to transition to pure rolling motion?

[WookieKx]

Homework Statement

A solid homogeneous cylinder of mass M and radius R is moving on a surface with a coefficient of kinetic friction μ_k. At t=0 the motion f the cylinder is purely translational with a velocity v₀ that is parallel to the surface and perpendicular to the central axis of the cylinder.
Determine the time t_R after which the cylinder performs pure rolling motion.

Homework Equations

None were given on the paper but I assume I'll be needing this:
When rolling: ƩE = K = 0.5mv² + 0.5I_cmω²

The Attempt at a Solution

Here's what I've worked through so far

When the cylinder is in pure translational motion:
ƩE = 0.5Mv₀² - f_kd
ƩE = 0.5Mv₀² -μ_knd

When the cylinder is in pure rolling motion:
ƩE = K = 0.5Mv_R² + 0.5I_cmω²

Due to conservation of energy:
0.5Mv₀² - μ_knd = 0.5Mv_R² + 0.5I_cmω²

As n = -Mg and I_cm = 0.5mr²

0.5Mv₀² + μ_kMgd = 0.5Mv_R² + 0.25MR²ω²

Simplifying:
0.5v₀² + 9.8μ_kd = 0.5v_R² + 0.25R²ω²

As v_cm = rω

0.5v₀² + 9.8μ_kd = 0.5v_R² + 0.25v_R²

0.5v₀² + 9.8μ_kd = 0.75v_R²

After this I can't really figure out how to go on apart from substituting d for:
0.5(v₀ + v_R)t
Which would add a time variable in.

This gives: 0.5v₀² + 4.9μ_k(v₀ + v_R)t = 0.75v_R²

The main things I'm stuck on are whether or not the v_cm at t=t₀ (v₀) and the v_cm at t=t_R (v_R) are the same or not and how to get rid of the coefficient of kinetic friction which I am not given a value for.

 

[ehild]

You need time so it is better to use equations where time is included.
The motion of the cylinder consist of translation of the CM and rotation about the CM. The force of friction decreases linear momentum and the torque of friction increases angular momentum. Write up both of them as functions of time and use the rolling condition v=Rω to find t_R when pure rolling is established.

ehild

 

[WookieKx]

So that would make something like:

Δp = Mv₀ - Mv_R = -f_kt

and

Δp = I_cmω = f_kRt

so

v_R = v₀ - μ_kgt

and

ω = (2μ_kgt)/R

so

v_R = 2μ_kgt = v₀ - μ_kgt

v₀ = 3μ_kgt

I always seem to have too any unknowns left over

 

[ehild]

Eliminate μ_kgt, you get V_R. Substitute back to get t_R in terms of the given data v₀, R, μ_k.

ehild

 

[WookieKx]

Ok so:

v₀ = 3/2*v_R
v_R = 2/3*v₀

But I don't think I can go on any further because I haven't actually been given any values for μ_k, v₀ or R

So the closest I can come is:

t = v₀/(3*μ_kg)
t = v₀/(29.4*μ_k)

 

[ehild]

But I don't think I can go on any further because I haven't actually been given any values for μ_k, v₀ or R

So the closest I can come is:

t = v₀/(3*μ_kg)
t = v₀/(29.4*μ_k)

OK, it is correct. No numerical data are given, you have to express the time with v_o, g and μ_k
.

ehild

 

[WookieKx]

Ok thanks for your help it was much appreciated.