Frictionless, Massless Problem from Hell!

My sister told me this physics problem, and I have no idea how to solve it. It goes like this. There are two identical balls, one lying a distance x from the edge of a frictionless table, and the other being held up in the air, a distance x from the edge of the table. (See diagram above.) They are connected by a massless rope.

At time t=0, the ball hanging in the air is dropped.

Do you see what’s going to happen? That ball will start to fall down and approach the table, while the ball sitting on the frictionless table will be pulled toward the edge.

Two questions, both of which I can’t answer:

(1) Which will reach the vertical plane of the edge of the table first? (In other words, will the ball on the frictionless table fall off the table before the falling ball hits the side of the table? Or vice versa? See my diagram below to see what the estimated path of the falling ball will look like. It curves towards the table, but also the distance from the corner of the table to the falling ball will increase as more time passes.)

(2) What is the equation for the path of the falling ball? (Again, see diagram below.)

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One comment

  1. Without putting pen to paper, I’m pretty sure I can answer the first question.

    If the rope connecting the two is inelastic and massless, both balls will experience a force of equal magnitude, albeit in different directions. The free ball also experiences acceleration due to gravity, which is always vertical. the ball on the table, being on a frictionless surface, can ignore the effects of gravity.

    The horizontal position of the balls will be the second integral with respect to time of the horizontal component of the force transmitted through the rope (gravity can be ignored, since it only acts in a vertical direction). For the ball on the table, this is always the full tension on the rope. For the free falling ball, this is somewhat less. Therefore, the integral and second integral for the free falling ball will be less, and the ball on the table will reach the edge first.

    Note that this doesn’t help you at all answer the second question or even anything else.

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