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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. enter image description here

RE60K
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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).

enter image description here

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...

Floris
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  • 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
  • I get the feeling from this answer that you do not understand the relationship between forces and acceleration.

    When the ball is in motion on a plank that causes friction on the ball, there is a net force present on the ball and it will *slow* down. Using Newton's 3rd law, the force of friction on the ball from the plank is equal to the force of friction on the plank from the ball. Therefore, were the plank not nailed down, it would move due to the friction from the rolling ball.

     – Dutch Brannigan Dec 28 '14 at 01:19
  • I understand the phrase "pure rolling" to mean "rolling without slipping bit without a force of friction retarding the sphere." If my assumption is wrong then so is my answer - but not because I don't understand the relationship between force and acceleration. – Floris Dec 28 '14 at 04:11
  • "Rolling is a combination motion in which an object rotates about an axis along a straight-line trajectory." Now I think where you are getting fudged up is that at the bottom of the rolling-without-slipping ball the velocity is zero. This does not mean the force is 0!!! Imagine throwing the ball straight up...when the ball reaches the zenith(the highest point) it stops momentarily before coming back down. Yet, the force of gravity was present the entire time. Anyway, the ball would not be able to roll without friction...it can slide! it can spin! but it cannot roll-without-slipping. – Dutch Brannigan Dec 28 '14 at 05:39
  • Thankyou both Floris and @DutchBrannigan. I well understand know, actually I knew it a year back but just forget everything, was therefore just revising evrything and you helped me a lot – RE60K Dec 28 '14 at 05:48
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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.

LSerni
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As I observe from the diagram the sphere is undergoing pure rolling,and as we know pure rolling does not need friction.So if the nail is removed there will be no force acting on the plank and hence the plank will remain at the same position and the sphere will continue the rolling motion.

Sagnik
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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.

Dutch Brannigan
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  • 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 04:13
  • 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 05:19
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    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
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    Please see the diagram I posted on http://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
  • Let us continue this discussion in chat. – Floris Dec 28 '14 at 19:50