A cylinder rolls without slipping

[athrun200]

Homework Statement

See the attachment Q5

Homework Equations

The Attempt at a Solution

Let's begin with part (a) first.
After finishing this homework (see the second attachment), I suddenly noticed 1 thing. The friction will gradually decrease to zero and then change the direction to become same as the motion direction.
Therefore, my work on part (a) would be wrong. Since I assume the friction is a constant (mgμsinθ).
How do I fix it?

And also, is my way to find the minimum coeff of kinetic friction correct?

And for part (b), it is really that easy?

 

[TSny]

For part (a), make sure you use the correct trig functions when finding the components of the weight. Otherwise, looks good!

For part (b), the friction is no longer kinetic friction when rolling without slipping.

 

[athrun200]

For part (a), make sure you use the correct trig functions when finding the components of the weight. Otherwise, looks good!

For part (b), the friction is no longer kinetic friction when rolling without slipping.

Is it possible to find out the static friction coefficient in this question? Or we can just let a new μ be static friction coefficient?

For part (a), do you mean that the friction would be either kinetic or static, there is not intermediate between them (i.e. the friction would increase or decrease from one to another)?

 

[TSny]

Part (a) looks good except for finding the correct expressions for the components of mg along the slope and perpendicular to the slope.

For part (b), you do not need a coefficient of friction. Just consider the torque and force equations and link them using the relation between linear and angular acceleration for rolling without slipping.

 

[athrun200]

Thank you! I will change the sin and cos!

 

[athrun200]

Oh, I found that the direction of the friction turns out to be to the left. Does it make sense?

 

[ehild]

Oh, I found that the direction of the friction turns out to be to the left. Does it make sense?

Before rolling, the cylinder shifts downward so the kinetic friction acts in the opposite direction, upward along the slope. With that direction, friction accelerates rotation and decelerates translation.

ehild

 

[TSny]

Oh, I found that the direction of the friction turns out to be to the left. Does it make sense?

Yes. There has to be a clockwise torque about the CM in order to give a clockwise angular acceleration as the cylinder accelerates down the slope.