bullet Woman Pushing on Three Blocks

 

Considering the situation presented in the figure where a woman is pushing three blocks at the same time, what makes each of the blocks move? For simplicity, let us assume that there is not friction between the blocks and the floor.

The woman applies a total force FEx to the three blocks. As a consequence of this force, the three blocks accelerate in the horizontal direction with the acceleration

The previous equation is obtained by applying the FBD technique to a single block of mass .

What is accelerating the block of mass m3? This block is pushed by the block of mass m2 with the force F32; which, in this kind of problems, is called the contact force . Since the acceleration of this block is known, the magnitude of the force can be calculated,

From the previous results, the magnitude of the force applied by the block of mass m1 over the block m2 can be calculated. Thus, over the mass m2 there are two forces acting on the horizontal direction, F23 which is the reaction force to F32, the push of block 2 on block 3, and the push of the block of mass m1 on the mass m2 . The net force acting on the block of mass m2 is

The acceleration of this block is also the same acceleration as the acceleration of the other blocks,

Equating the last two relations, an equation for F21 is obtained,

Thus,

The same analysis can be made for the block of mass m1  . In this case, the forces acting on the block are the external force and the force exerted by the second block on the first block. This last force is the reaction to the push of the block of mass m1 on the block of mass m2. The net force acting on the mass m1 is

Newton's second law applied to the first block is

 .

Thus,

The previous relation is the same equation as Newton's second law applied to the three blocks when considered as a single mass.

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by Luis F. Sez, Ph. D.    Comments and Suggestions: LSaez@dallaswinwin.com