# Two Balls - One Suspended, One Stationary

## Homework Statement

Two small balls of the same mass and shape are attached by a massless string of length L. Ball A is placed on a frictionless table and ball B is held a distance L/2 over the edge with the string taut.

When ball B is dropped, will it hit the side of the table before A reaches the edge or will A reach the edge first?

You do not need to write any equations for this question, just explaining your reasoning will suffice.

None given.

## The Attempt at a Solution

I was thinking that this problem had something to do with a pendulum, as when B is swinging down about the point where the string leaves the table, it moves so in a "pendulum-like" fashion. Maybe it has something to do with angular momentum.

Honestly, the problem could also probably be solved with just a simply free-body diagram utilizing Newton's Second and Third Laws, but I just don't know how to approach this.

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Simon Bridge
Homework Helper
The amount of string over the edge of the table is r(t): r(0)=L/2.

At t=0, B starts accelerating downwards, pulling A and making r longer. The tension on the string changes the motion of B into an arc.

This is the sort of thing that is done rigorously with hamiltonian mechanics, but you should be able to reason it out from first principles.

note: if r(0)~0, the B hits the wall first, if r(0)~L then A falls off the edge first.
Is r(0)=L/2 the sweet-spot where they hit at the same time?

Sorry, I have no idea how to use Hamiltonian mechanics.

I am enrolled in a very basic introductory mechanics class so is there anything I could use from good old Newtonian mechanics?

Bump.

I would appreciate any response, as I need to finish this problem soon.

jedishrfu
Mentor
why not actually do the experiment and video tape so you can slow it down.

or you can do the experiment by still holding onto Ball B and as you feel tension on the string move to keep it to a minimum. You should notice that as ball B is "falling" it pulls at A and gets a little closer to the table. Also because Ball B is "falling" it is going faster but is being held back by the drag of Ball A.

Also maybe do the experiment with two coins like quarters with a thread attached between them. This will slow down the motion a little so you can then understand what will happen.

I need a theoretical, not an experimental, explanation of this system.

Is there any "theroetical" approach I can take? I am tempted to say that it takes the same amount of time for A to fall off as for B to hit the side of the table, but I don't have strong justifications to back that claim up...

Simon Bridge
Homework Helper
Well sure, you said so yourself - draw a free-body diagram for each mass, write the relations for the forces. For B, the angle of the tension force will change during the motion. You want to relate this to how far away the side of the table is - because B's motion in that direction is what you care about.

If B has fallen distance y, then it is x away from the table and the length of the string from B to the table is r. If the angle the string makes to the horizontal is Q, then sinQ=y/r and cosQ=x/r

The acceleration of A is given by a=T/m, which is also tells you how r increases with time... is there constant acceleration though? What determines the acceleration of B in the x and y directions? How are these related to the acceleration of A?

As the question states though, you don't have to do this.
Use your understanding of how acceleration works.

Don't knock the experiment - it will guide you to the theoretical explanation you need.
You are not required to do the math - you are supposed to be demonstrating your understanding.
You are stuck because you don't understand the system - experiment is how you gain understanding.

I'm sorry you are working to a deadline - can't help you there: understanding does not come quickly.)

Will A fall off first because it accelerates at some a, while B accelerates in the x-direction at some value a_b < a, because gravity is split up into lesser components??

Simon Bridge
Homework Helper
That's the kind of thinking that the problem is looking for.

The total acceleration of A and of B must be the same (a) at all times - because the string attaches them. Thus the x component of the acceleration of B will always be less than a (re. Pythagoras). They each have the same distance to cover. I'd go with that if I were you. Mind you, I cannot guarantee that's the right answer - with these types of questions it's seldom about getting it right and more about making your case anyway ... and I'm not the one marking it.

If you have a bit more time you can try to flesh it out.

The sort of thinking is an important part of science - particularly physics and angineering.
Suggest practice. Books like Gedanken Physics are useful for this.

(I know that's not terribly reassuring - it's just that I cannot do the problem for you. I am deliberately not doing the problem so I cannot accidentally give the game away.)