Cylinder Rolling Down an Incline Without Slipping

[Leo Liu]

Homework Statement

This is a conceptual question.

Relevant Equations

$\tau = I\alpha$
$F=ma$
$f=\mu F_N$

First we let the static friction coefficient of a solid cylinder (rigid) be $\mu_s$ (large) and the cylinder roll down the incline (rigid) without slipping as shown below, where f is the friction force:

In this case, $mg\sin(\theta)$ is less than $F_{max}$, where $F_{CM,max}$ is the maximum force (excludes the friction, yet includes the the x component of the gravity ) can be exerted on the cylinder without causing it to slip.

My questions are:

1. How do we calculate the force of friction? Is it equal to $mg\ cos(\theta) \mu_s$ (which is the same as $f=\frac{1}{2}ma_{CM,max}$) or $f=\frac{1}{2}ma_{CM}$? (please note that the $a$ in the latter is a variable)

2. What is the norm of translational acceleration $a$ in this example?

3. If the cylinder does slip when rolling down, how would you find the force of friction? Do you need any additional information?

Derivation of the formula in question 1:
$$\tau = I\alpha$$
$$Rf\sin(90) = \frac{1}{2}mR^2 \frac{d\omega}{dt}$$
$$f=\frac{1}{2}m\frac{d\omega R}{dt}=\frac{1}{2}m\frac{dv_{tan}}{dt}=\frac{1}{2}ma_{tan}=\frac{1}{2}ma_{CM}$$

 

[kuruman]

Homework Statement:: This is a conceptual question.
Relevant Equations:: $\tau = I\alpha$
$F=ma$
$f=\mu F_N$

First we let the static friction coefficient of a solid cylinder (rigid) be $\mu_s$ (large) and the cylinder roll down the incline (rigid) without slipping as shown below, where f is the friction force:
https://www.physicsforums.com/attachments/259352
There must exist a force $F_{max}$ pointing down the incline, which ensures that the cylinder rolls without slipping. In this case, $mg\sin(\theta)$ is less than $F_{max}$.

Why must there be such a force? Complete your free body diagram (FBD).

My questions are:

1. How do we calculate the force of friction? Is it equal to $mg\ cos(\theta) \mu_s$ (which is the same as $f=\frac{1}{2}ma_{CM,max}$) or $f=\frac{1}{2}ma_{CM}$? (please note that the $a$ in the latter is a variable)

Complete the FBD and write Newton's 2nd law for translations and rotations, this means 2 separate equations.

2. What is the norm of translational acceleration $a$ in this example?

It is what comes out of the 2 equations.

3. If the cylinder does slip when rolling down, how would you find the force of friction? Do you need any additional information?

You will need the coefficient of kinetic friction.

 

[Leo Liu]

Why must there be such a force? Complete your free body diagram (FBD).

Hi, I have updated my diagram and improved the clarity of my question.

this means 2 separate equations.

Sorry, I don't quite understand this point since I think the two types of motion are linked together, would you care to explain?.
My understanding:
$$\because x_{arc} = x_{CM}$$
$$\therefore R \omega = v_{CM}$$
$$\therefore a_{tan} = a_{CM}$$
This is how I got the expression of $f$ in terms of $a_{CM}$, but I don't know if it is right.

 

[kuruman]

Hi, I have updated my diagram and improved the clarity of my question.

Sorry, I don't quite understand this point since I think the two types of motion are linked together, would you care to explain?.
My understanding:
$$\because dx_{arc} = dx_{CM}$$
$$\therefore dv_{tan} = dv_{CM}$$
$$\therefore da_{tan} = da_{CM}$$
This is how I got the expression of $f$ in terms of $a_{CM}$, but I don't know if it is right.

What do these three equations mean? What happened to Newton's 2nd law? Your FBD is still incorrect. It shows the same force (weight) twice; once as a single vector straight down and then again as its components perpendicular and parallel to the incline.

 

[haruspex]

1. How do we calculate the force of friction? Is it equal to $mg\ cos(\theta) \mu_s$

Only if it is on the verge of slipping. Rolling just means the frictional force does not exceed $N \mu_s$.

2. What is the norm of translational acceleration a in this example?

"Norm"? I don't understand the question.

 

[Leo Liu]

What do these three equations mean? What happened to Newton's 2nd law? Your FBD is still incorrect. It shows the same force (weight) twice; once as a single vector straight down and then again as its components perpendicular and parallel to the incline.

I just intended to show that the magnitude of the tangential acceleration at the point where the cylinder touches the slope is equal to the translational acceleration of the cylinder, as its direction is opposite to the latter. In the FBD, I did not want to remove the gravity so I decided to add red colour to its component.

 

[Leo Liu]

Only if it is on the verge of slipping. Rolling just means the frictional force does not exceed $N \mu_s$.

"Norm"? I don't understand the question.

Does that mean the magnitude of the force of friction $f = \frac{1}{2}ma_{CM}$? I meant "magnitude" when saying "norm". Thank you.
Also, @kuruman mentions that kinetic friction will come into play when the cylinder slips. If this is the case, what are the uses of $\mu_s$ other than finding $F_{max}$?

 

[haruspex]

Does that mean the magnitude of the force of friction $f = \frac{1}{2}ma_{CM}$?

Yes.

Also, @kuruman mentions that kinetic friction will come into play when the cylinder slips. If this is the case, what are the uses of $\mu_s$ other than finding $F_{max}$?

None.