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A (really) small problem

Here's a quick one.

Refer to the accompanying figure.

Figure to accompany a really small problem

 

It shows a frictionless glass tube lying on a horizontal desktop. The tube is smoothly bent at a couple of places as shown in the figure.

Suppose that a small steel ball (say, one taken from a ball-bearing) enters the tube at the point A, with an initial velocity of v.

The problem is to predict the local speeds of the ball as existing at the following points/sections. 

(i) at the point C (the tangential speed)
(ii) at the point F (the tangential speed)
(iii) in the straightline section GH

For each point or section, it would be enough to predict whether the local speed there would be greater/smaller/equal to the initial value v.

If the problem cannot be solved as stated, feel free to say so.

In any case, it is important that you are able to provide brief reasons for your answers.

Comments

This is an old post, but still considering no one responded (?)......

assuming:

1. the portion AB is straight and the velocity vector v of the ball is along the vector AB, and

2. if the diameter of the tube is the same as the diameter of the steel ball AND

3. if the tube is in the horizontal plane (no gravity) and frictionless (so no rolling, no frictional/heat losses) AND empty and

4. The tube is smoothly bent (no impact, no sound losses) and

5. the steel ball is considered rigid (no deformation)and

6. No body forces exist apart from gravity in the region under consideration

then obviously all the speeds are going to be the sameas the initial value |v|. Why?

If any of these are violated, we will need more information to solve the problem.

Dear Ms. Dude,

Thank you for amplifying the problem. Now, I would like to see what you have to say about the solution.

 

Temesgen Markos's picture

Hi Ajit,

If there is no friction or other energy loss then the total energy (kinetic + potential) remains the same. In your case, the glass tube is horizontal. If that means the motion of the ball is also on a horizontal plane then the potential energy remains the same, which mean the kinetic energy of the ball stays constant, i.e. the magnitude of the velocity does not change.

If you have a motion constrained to a smooth curve, then the velocity is always tangent to the curve. 

With that we conclude the tangential velocity at every point along the path is the same as the initial value. 

Nothing says that the tube is fixed to the desktop.  An unconstrained frictionless tube lying on a horizontal desktop will move once the ball hits the first curve, and since the tube is frictionless, it will keep moving...off the edge of the desktop and into a rotating freefall if gravity is present.  All things considered, the ball might not even reach the other end.

Good catch lol. But even if the tube isn't constrained to stay on the desktop, we should still be able to conserve linear/angular momentum and if absolutely no dissipative forces exist, we shd also be able t conserve kinetic energy and actually obtain a solution, assuming the ball moves like a particle and we know properties of tube and equations of curves.

In any case, center of mass always moves along vector AB and has known speed.

 

 

Temesgen Markos's picture

In Cooper's case where the glass tube can move in the horizontal plane, the speed of the ball will be reduced once the glass tube starts moving, unless there is some source of energy.

 

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