# Two masses connected by a string in horizontal frictionless table

Suppose that two masses with different mass are connected by a string in horizontal frictionless table. Then when there is equal acceleration for two masses, string force would extend in some direction. My question here, why don't two boxes create canceling effect for string's stretching? (Often, in we just calculate one box's force opposite to the direction of acceleration and equals to kx, which solves the problem. But why?)

• is the common acceleration of each of the masses in the same direction? Sep 19 '13 at 7:38
• yes. I said there is equal acceleration. Sep 19 '13 at 8:03
• Are there external forces on both masses to cause the acceleration, or is there an external force on one mass which is pulling the spring which is pulling the other mass? Sep 20 '13 at 15:35

If I understood your question correctly, you are asking why does the spring stretch when it is attached from both sides to two different masses having the same acceleration moving in the same direction.

The reason is that there is a net force exerted on the spring. Newton’s second law states that the force is equal to the mass multiplied by the acceleration. Since the two bodies have different masses and same acceleration, that means one of them is exerting a larger force on spring than the other. This net force causes the string to stretch/compress depending whether the leading mass or the lagging mass has a larger mass.

• Are you assuming that the force on the spring from a given mass is the same as the force experienced by that mass? Perhaps we're interpreting the question differently -- I interpreted the acceleration to be caused by an external force on the block (which would not equal the spring force for non-zero acceleration). If all the forces are internal to the spring-block system, the acceleration of the system is zero. Sep 20 '13 at 16:17
• @TimGoodman : The answer is absolutely correct, What happens is that after application of forces on different blocks, the spring adjusts itself to equate forces on each side, internal forces cancel out after this configuration has been achieved and the whole system starts moving like a single body ! Jan 18 '14 at 20:54

If you're talking about someone pulling on just one of the blocks, then the spring stretches because it exerts force to pull the other block.

Consider two blocks with masses M and m, joined by a spring. Suppose I exert a force F on mass M, this will cause it to accelerate, and thus will stretch the spring. As the spring stretches, it exerts a force on the second block (as well as on the first block), causing the second block to accelerate as well.

The spring will keep stretching until the accelerations on the two blocks are equal. At that point one block feels a force f from the spring, and the other feels a net force F - f. Since the accelerations are equal, we know (F - f) / M = f / m, and can use this to calculate the spring force (and how much it stretches, if we know the spring coefficient k.)

If on the other hand you were to put external forces on both blocks so as to give them equal accelerations, then the spring wouldn't need to stretch at all.

• For an idealized spring of negligible mass, at any rate. But the general idea holds regardless. Sep 20 '13 at 16:13