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.

 

[baba89]

To calculate the distance the cylinder will pass until it starts rolling without sliding, we can use the principle of conservation of angular momentum. Initially, the cylinder has no angular momentum as it is not rotating. When it starts sliding, it will gain linear momentum and hence angular momentum. However, due to the friction coefficient, the cylinder will experience a torque that will oppose its motion and eventually cause it to start rolling.

The key to solving this problem is to find the point where the torque due to friction becomes equal to the torque due to the cylinder's angular momentum. This is the point where the cylinder will start rolling.

First, we need to calculate the angular momentum of the cylinder when it starts sliding. This can be done by using the formula L = Iω, where L is angular momentum, I is the moment of inertia, and ω is the angular velocity. Since the cylinder is not rotating initially, ω = 0. Therefore, the angular momentum at this point is also 0.

Next, we need to calculate the torque due to friction. This can be done by using the formula τ = µmgR, where µ is the friction coefficient, m is the mass of the cylinder, g is the acceleration due to gravity, and R is the radius of the cylinder. This torque will act in the opposite direction to the motion of the cylinder.

Now, we can equate the torque due to friction to the angular momentum to find the point where the cylinder will start rolling. This can be expressed as µmgR = Iω. Rearranging the equation, we get ω = µmgR/I.

Finally, we can use the formula v = ωR to find the linear velocity at which the cylinder will start rolling. This can be expressed as V0 = ωR. Solving for R, we get R = V0/ω.

To calculate the distance the cylinder will pass until it starts rolling, we need to find the time it takes for the cylinder to reach this point. This can be calculated using the formula t = V0/a, where a is the acceleration of the cylinder. Since the cylinder is initially moving with a constant velocity, the acceleration at this point is 0. Therefore, the time taken to reach this point is infinite.

In conclusion, the distance the cylinder will pass until it starts rolling without sliding is infinite. This is because the time taken to reach this point is infinite, as the cylinder needs to overcome