What will happen if I remove a nail which stops a plank from moving on a smooth floor because of a solid sphere pure rolling on it?

What will happen if I remove a nail which stops a plank from moving on a smooth floor because of a solid sphere pure rolling on it?The plank has it's upper surface which is in contact with the sphere rough, therefore the sphere is rolling(purely). I think if I remove the nail then the plank can also move, which is mainly due to absence of any external force, but don't have an analytic idea, you may call it intuition.

• A simple diagram could help i feel for people to have any intuition. – Gowtham Dec 26 '14 at 17:34
• @Gowtham does it help now – RE60K Dec 26 '14 at 17:41
• Now turn it into a free-body diagram, including 3rd law action-reaction force pairs. If the sphere is rolling freely without slipping, then the contact point must be stationary with respect to the plank, i.e., no movement, so there is a static frictional force there. – lionelbrits Dec 26 '14 at 17:53
• This link may be of help. – jayann Dec 26 '14 at 18:45

If the sphere is already rolling when you remove the nail, there will be no net force on the plank and it won't matter whether the nail is there or not.

For the plank to move, it needs to experience a net force. For example, if the sphere was the wheel of a car that's accelerating, then as the wheel accelerates to the right, the plank will accelerate to the left. But in the steady state, nothing will happen to the plank as there is no net force on it.

UPDATE if the ball experiences friction w.r.t. the plank, the situation is a bit more interesting / complex. We know that friction must result in the slowing down of the ball, but there's a counter-intuitive thing going on here: if I supply a horizontal force to the ball that's pointing to the left at the contact point of ball and plank, then that would imply an accelerating torque on the ball... We solve this by looking closely at this diagram which is a close-up of the imagined point of contact and which explains one mechanism of rolling friction (where the flat surface deforms; it's possible to do the same thing when the sphere distorts but it's intuitively harder).

The "zoomed in" view (lower half of the picture) shows that there is some distortion that results in a net vertical force that is offset with respect to the force of gravity on the center of the sphere - this results in a retarding torque. As the ball slows down, the plank speeds up (because of the horizontal component $F_h$) - this is of course necessary to preserve momentum in the horizontal direction.

Quite a detailed description of this can be found at http://askthephysicist.com/classical%20mechanics.html - it's a very long page, so to find the relevant article you will want to find the phrase "My question has to do with traction and the movement of a wheel" which is the start of the snippet on rolling friction etc.

I would like to acknowledge the helpful discussion between myself and Dutch Brannigan which made me decide to add a diagram and further explanation...

• but the friction which supports the rotational motion, is there too, isn't frictional force an action-reaction pair? – RE60K Dec 26 '14 at 18:11
• There is no net force (horizontally) to keep the sphere rolling - if there were a force (because the sphere is either accelerating or decelerating) then there would be a reaction. But as described by you there is no acceleration, so no force. – Floris Dec 26 '14 at 18:23
• Net force doesn't keep a ball moving. See Newton's 1st Law of Motion. – Dutch Brannigan Dec 26 '14 at 20:55
• what about the centre of mass being at rest in the absence of external force? – RE60K Dec 27 '14 at 18:05
• It doesn't have to be at rest - it can be moving at a constant speed... – Floris Dec 27 '14 at 19:26

From a qualitative point of view:

If there is friction between the ball and the plank, the ball will eventually stop relative to the plank.

When this happens, the system ball + plank must still possess the same momentum that initially was invested in the ball alone, which means that the system must be moving, relative to the frictionless floor, in the same direction that the ball was rolling in at the beginning.

Since the ball's slowing down is gradual, the plank's acceleration is also gradual.

When the nail is in, it transmits momentum from the plank to the floor, and the ball moves right while the planet moves imperceptibly in the opposite direction.

The plank would start to move in the opposite direction of the ball. A simple free-body diagram will show that. Now, how far/fast the plank will move will depend on its mass, the mass of the ball, the coefficient of friction between the two, and the gravity present on whatever world this thought experiment is done on. Also the length of the board would come into play, as the friction is slowing down the ball...given a long enough board the ball will eventually come to a stop, and when it does, the plank will also have stopped.

• If friction is occurring between ball and plank, wouldn't the plank start moving in the direction of the ball? – Floris Dec 28 '14 at 4:13
• Newton's 3rd law of motion: Every action has an equal and opposite reaction. It's like a hammer and a nail. You hit the nail and the nail feels a downward force. Simultaneously the hammer will feel an upward force of the same magnitude as the nail felt. – Dutch Brannigan Dec 28 '14 at 5:19
• Let's say the ball is moving to the right. Friction slows the ball down, then force of friction as experienced by the ball must be to the left. The reaction to this is a force experienced by the plank, and pointing to the right. The plank would start to move in the direction of the ball, and not "in the opposite direction of the ball". – Floris Dec 28 '14 at 14:13
• The plank would start moving in the direction of the ball? Do you know what a free-body diagram is? In fact, what is your background, Floris? You are speaking pretty confidently for someone who doesn't understand Newton's 3 laws of motion. – Dutch Brannigan Dec 28 '14 at 19:17
• Please see the diagram I posted on imgur.com/sxKALxk . This shows how friction will result in a force on the plank in the direction of the ball. Conservation of momentum demands it: if the ball slows down, the plank speeds up. I continue to believe you are thinking of a different scenario, or you wouldn't be arguing so forcefully - so please look at my diagram and see if we can agree. I apologize if I sound confident. I don't think I need to defend my credentials. – Floris Dec 28 '14 at 19:40