How Far Will a Sliding Cylinder Travel Before It Rolls Without Sliding?

[giladbr]

A full cylinder is placed on a straight plane with a friction coefficient µ (static=dinamic). The cylinder is hit in the middle and begins sliding without rolling in a linear velocity V0. The acceleration of gravity is g.

Calculate the distance the cylinder will pass until it starts rolling without sliding.

Thanks in advance!

 

[Chrisas]

As with all problems like this, set up a coordinate frame, draw in the force vectors, then apply Force = ma and Torque = I * (w_dot).

This gives you three scalar equations. If you put the coordinate frame so that y is normal to the ramp and x is down the ramp, then y_double_dot is zero and you only have to deal with x_double_dot, and w_dot.

Setting y_double_dot = 0 gives you the normal force and thus the friction force in terms of parameters you know (mass, theta, g, u). You know all the initial conditions, so you can integrate to find both x(t) and w(t).

If only we knew what t was, we would be done. You would plug t into x and solve. You still have one condition you haven't used yet. We want the time t when pure rolling occurs. What is the relation between w and x_dot for pure rolling? Plug in w and x_dot that you found into this condition and solve for t when rolling occurs.