# Notes on the Frictional Forces of Rolling Objects

### by Ian G. Zacharia

Dear Students,

I am sending this email before your test on Friday to help you on a potentially confusing topic: the role of friction for an object that is rolling without slipping. For those of you in my early Tuesday sections, this is especially relevant as I may have mispoke on some of these things. This email is long, so I will bracket sections that you can just skim.

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First I highly recommend reading p. 302 in Young and Freedman which helps give a good physical intuition for what's going on in a wheel that is "rolling without slipping". The two big takeaways from this section (which have also been mentioned in lecture) are

1. Vcm/R = w (I use w in this text to represent omega, the angular velocity) is the condition for rolling without slipping (if the surface the wheel is on is not moving). If w were larger the wheel would spin too fast like when your car is on an icy road and the wheel slips. If w is smaller, the wheel isn't yet spinning fast enough like when an airplane first lands and the wheel skids until kinetic friction increases w up to Vcm/R.

2. The contact point of the wheel has ZERO velocity so whenever we deal with friction of a wheel that is rolling without slipping we are dealing with static friction (as mentioned above, if the wheel is skidding the contact does have a velocity so it is kinetic friction). ]

Second, a big point of confusion for me when looking at these problems was to figure out which way static friction is pointing. It seemed each problem had a different direction! I will show some examples momentarily, but let me describe two methods that I think will help you figure out which way friction should point.

Method 1:
Think about the net acceleration and net angular acceleration of the object and deduce which way friction must point to give the resultant accelerations (after taking into account other forces in the problem which may also cause these accelerations).
Method 2:
Think about what would happen in the absence of friction versus what is happening with friction and from the difference deduce which way friction is pointing.

[Both methods should work equally well, but its good to do both to make sure you have it right. Now let's look at 3 examples from the problem set.]

### EXAMPLE 1 - Accelerating Car (10.7)

NOTE: The answer to the hint on pg. 338 draws the direction of friction in the wrong direction! See solution set for the correct direction.

In this problem we have a car accelerating, let's say from my perspective it's accelerating to the right. [ Then the torque on the axle must be clockwise to spin the wheel in the right direction. But which way does the static friction on the contact point of the wheel point (assuming rolling without slipping)? ]

Method 1:
There is a net acceleration to the right, and the only horizontal force on the wheel is friction thus is must point to the right.
Method 2:
If there were no friction (closely approximated by a car trying to drive on ice) the wheel would spin harmlessly and the car would not move forward. If there is friction, a point at the bottom of the wheel would be driven by the axle to the left. Thus the axle exerts a leftward force on the point at the bottom of the wheel, and friction will oppose this force and act to the right. Equivalently, the bottom of the wheel tends to slip to the left, so friction pushes it to the right.

### EXAMPLE 2 - WHEEL ON INCLINED PLANE (10.12)

Consider a wheel rolling down an inclined plane. [The forces acting on it are gravity (acts at the center of mass for purposes of computing net torque due to gravity), the normal force at the contact point and friction at the contact point. Does friction point up or down the inclined plane? (assume rolling without slipping)]

Method 1:
Since the wheel is accelerating, and angular acceleration (about CM) equals a/R, it has a net angular acceleration (about the CM). The only possible force that could cause a torque (to create this angular acceleration) about the center of mass is the force of friction, and since the wheel is rolling down the plane the force must be up the plane to cause the right torque (see diagram on p. 335).

Method 2:
In the absence of friction (and thus torques) the wheel would just slide down the incline and since instead it also has a rotation we choose friction to be in the direction up the plane to cause the proper rotation. [ Or I could have said the gravity tends to push the wheel down the incline (as if it were just sliding down) at the contact point so friction opposes that push and points up the incline. ]

### EXAMPLE 3 - Airplane Landing 9D.6 (solns p.7)

I actually just want to consider the wheel of the airplane after it has stopped skidding and is rolling without slipping. [ Remember to get to that point, the airplane's wheels upon impact at first have velocity Vo but no w. The wheels skid, and kinetic friction opposes the direction of motion and causes the wheel to spin faster and faster until w = Vcm/R. Why does kinetic friction stop here? Because at this point remember, the velocity of the contact point is zero so we could only have static friction. ]

Now, we basically just have a wheel at some velocity Vcm and the correct angular velocity for rolling without slipping (w=Vcm/R). A few seconds later, what does the force diagram look like? In what direction does friction point?

Answer: There is no static friction!

Method 1:
The velocity and angular velocity don't change anymore, so if there were any horizontal friction the wheel would be constantly speeding up or slowing down which it is not!
Method 2:
If we took this wheel with Vcm and w=Vcm/R and put it on a frictionless surface what would happen? Well, there would be no forces to change the velocity or angular velocity, so it would continue to roll without slipping, same thing it does when there is friction. Thus friction plays no role.
[Now, you might stop here and say "Wait a second, are you telling me that if I rolled a ball down a real infinite corridor it would roll forever?" According to what I have just said, yes. So our situation IS somewhat idealized even though we are seemingly taking friction into account. What's missing? Another kind of friction called "rolling friction" which is very small (see pages 137, 304 ex 10-7, and 306-7 in Young and Freedman). It is caused by deformations in the surface being rolled over (e.g. the ground) and the rim of the wheel (like in your bike tire). We have neglected this in the problems here and unless otherwise specified you may neglect it when solving 8.01 problems.]

Well, if you've read this far, I hope all of this helped. Please don't hesitate to email me with any questions about this topic or the rest of the course. Best of luck on the exam!