Rolling cylinder on surface with friction (initially slipping)

[vraeleragon]

Hi, I'm trying to solve this, but I keep getting both sides of the equation to be the same >.<
A solid cylinder is initially moving along a flat surface without rotating. Due to the action of friction, it eventually begins rolling without slipping. The coefficient of friction is 0.22, the radius of the cylinder is 0.5 m, its mass is 2.7 kg and its initial center of mass velocity is 1.5 m/s

Questions: a.) How long does it take to reach a rolling without slipping condition? b.) What is the total energy change of the system between the initial condition and the establishment of rolling without slipping? c.) Will it still roll without slip on an incline?

I tried using FΔt=mΔv and d=v$_{0}$t+1/2at$^{2}$so i got the distance for slipping d=(v$_{i}$$^{2}$-v$_{f}$$^{2}$)/(2μg)
And then I did 1/2mv$_{i}$$^{2}$ = fd + 1/2mv$_{f}$$^{2}$ + 1/2Iω$^{2}$ I kept getting the same answer no matter what formula or approach I use. I have no clue how to do this. Please help. Thank you :)

 

[tiny-tim]

hi vraeleragon!

I tried using FΔt=mΔv and

no, it starts at slipping without rolling, goes though slipping with rolling, and ends with rolling without slipping

use the work energy theorem …

work done = friction "dot" displacement ​

 

[ehild]

The force of friction decelerates translation of the CM and the torque of friction (with respect to the CM) accelerates rotation. Write up both equations and solve for the linear velocity v and angular velocity ω in terms of time. Find t when the condition of pure rolling applies (do you know it?).

ehild

 

[vraeleragon]

oh i forgot to say that i solved it. thank you.

 

[asteriatic]

I would suggest looking into the equations and variables used in your calculations. It is possible that you may have made a mistake in your algebra or used incorrect values for the variables. Double check your work and make sure you are using the correct equations for the given scenario.

Additionally, to solve for the time it takes to reach rolling without slipping, you can use the equation v = ωr, where v is the linear velocity, ω is the angular velocity, and r is the radius of the cylinder. You can also use the equation a = αr, where a is the linear acceleration and α is the angular acceleration. With these equations, you should be able to solve for the time it takes for the cylinder to reach rolling without slipping.

For the total energy change, you can use the equation E = KE + PE, where KE is the kinetic energy and PE is the potential energy. Initially, the cylinder only has kinetic energy, and once it starts rolling without slipping, it will have both kinetic and potential energy. You can solve for the energy change by subtracting the initial kinetic energy from the final total energy.

Lastly, for the incline question, the cylinder will still roll without slipping as long as the force of friction is enough to counteract the component of the cylinder's weight that is parallel to the incline. You can use the same equations mentioned above to solve for the force of friction and determine if it is enough to keep the cylinder rolling without slipping.