# What role does static friction force play for a rolling object? How can I know what direction it points?

Fundamentally, what does a friction force do for a rolling object? I am very confused by the way my textbook explains it. In my textbook , it says:

A wheel rolling on a horizontal flat surface at a constant velocity experiences no friction force. Why?

On the same surface, there is an acceleration of the wheel pointing to the right (probably caused by a force), so the ball is angularly accelerating in the clockwise direction. In this case, a friction force appears, and it is also pointing to the right. How come? What does the friction do for this wheel?

On an inclined plane, a ball freely rolls down the surface. The direction of friction is up the ramp, which confuses me because in the previous example the friction force was in the direction of the wheel's acceleration.

And there is a difference if a wheel is freely rolling and if there is a torque acting on the ball's center of mass. WHY???

What does friction do in these cases? How does it cause an object to roll?

• Static friction, as it acts in rolling without slipping. – Richard Song Jan 11 '15 at 18:41
• And static friction supposedly prevents the wheel from sliding, right? – Richard Song Jan 11 '15 at 18:46
• Then how do I know which way something would slide? How do I know which way a wheel WOULD slide? – Richard Song Jan 11 '15 at 18:58
• – dmckee Jan 11 '15 at 22:25

## 2 Answers

Hopefully this will answer at least some of your questions. If you consider a ball intially at rest on a frictionless surface, if a force is exerted through the centre of mass of the ball it will slide across the surface with no rotation, there will only be translational motion.

If you consider the case where there is friction, if the force is again applied to a stationary ball the frictional force will act in the opposite direction to the force but at the edge of the ball that rests on the ground. This friction applies a torque to the ball which starts the rotation. So static friction is infact necessary for a ball to begin rolling.

The condition for rolling at speed $v$ is that the angular velocity of the ball is given by $$v=\omega r$$ from this the top of the ball will move at speed $2v$ the centre of mass of the ball will move at $v$ and the bottom edge of the ball will instantaneously be at rest. So as the edge touching the ground is stationary it experiences no friction.

So friction is necessary for a ball to start rolling but once the rolling condition has been met the ball experiences no friction.

• Thank you for your explanations. I have concluded the following: please correct me if I am wrong. – Richard Song Jan 11 '15 at 19:59
• If a ball has a force acting through its center of mass (and therefore no torque is generated by that force), with no friction it will just slide forward. If the force acted towards the right, then there would be a friction force acting towards the left, thus generating a torque about the center of mass. – Richard Song Jan 11 '15 at 20:02
• In regard to Chris2807's answer, I have a question: If a ball is rolling down the hill on its own, there is a continual component of gravitational force that pulls it down the incline. Since this force produces no torque about the center of mass, I have concluded that a frictional force must be exerted on the contact point at all times when the ball slides down the incline, or else it would slide. What do you mean by the fact that once the rolling condition is met the ball experiences no friction? – Richard Song Jan 11 '15 at 20:17
• Nice. You can also take the opposite approach: consider what happens if an external couple (pure torque) is supplied around a horizontal axis. Without friction the ball spins in place, but with friction the ball applies a force to the ground and the reaction force both cancels much of the applied couple and causes the ball to begin translational motion. – dmckee Jan 11 '15 at 22:23
• @RichardSong once the ball has started to roll at an angular velicty $\omega=\frac{v}{r}$ it will continue to roll forever assuming we don't have air resistance etc as there will be no friction with the ground. For the inclined plane the key is that the ball is constantly accelerating due to gravity and so its translational speed keeps increasing therefore the rotational speed must also increase, this increase in rotational speed comes from friction with the plane. In a very loose sense it is like $\omega r$ is trying to constantly "catch up" to $v$. – Chris2807 Jan 11 '15 at 23:48
1. If the wheel is rolling at constant velocity without any other forces acting, then there is no tendency for slippage between the wheel and the surface, so there is no friction. Friction will only act to try to prevent slippage between 2 surfaces.

2. Presumably in this case there is a torque acting on the axis of the ball/wheel? This torque will act to give angular acceleration to the wheel (clockwise). If there was no friction then the angular velocity would increase and there would be slippage between the wheel and the surface. However, friction exists and will try to prevent the slippage, which is why it acts to the right. In trying to prevent this slippage, the friction also exerts a horizontal linear force on the wheel, giving it acceleration to the right, so it speeds up. This is how a car accelerates horizontally, using friction between the wheels and the road.

3. On the inclined plane, the ball will tend to slide downwards and if there was no friction then again there would be slippage between the ball and the inclined surface. Perhaps the best way to think about it is: "Which way is slippage going to occur if there was no friction?" Friction will always try to counteract slippage, so here it acts up the slope to try to prevent the ball slipping down the slope. This has the effect of applying a moment to the ball, which will cause it to start rolling.

• So what if there is a force that causes no torque and a force that causes torque that act on a wheel. So for example, one force acts on the center of mass while another acts on the top of the wheel. The wheel is on a flat surface. Does the wheel need friction to roll? – Richard Song Jan 11 '15 at 21:29
• For the force through the center, the wheel will need friction to start rolling, otherwise it will just slide horizontally. For the force on the top, if there is no friction then the wheel will start both moving laterally and rotating. The subsequent motion will depend on how the force behaves as the wheel turns - whether it turns with the wheel or stays pointing horizontally (I assume it acts at the same point). – Time4Tea Jan 11 '15 at 21:56