MASSACHUSETTS INSTITUTE OF TECHNOLOGY
DEPARTMENT OF PHYSICS
8.01 Fall 1998
Notes on the Frictional Forces of Rolling Objects
by Ian G. Zacharia
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.
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
- 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.
- 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.
[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
- 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.
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
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!
[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.]
- 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.
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!
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Last modified: Wednesday, November 18, 1998 10:58 pm