11 November 2022

Mail Time: how do you rank the forces on a block dropped onto a spring?

I was asked about a ranking task in which block 1 is dropped onto block 2, which is attached to a vertical spring.  The blocks compress the spring, and come briefly to rest in the position shown in the right-hand diagram:

We're asked to rank the (magnitude of) three forces at this instant of maximum spring compression:
  • Fa is the upward force of block 2 on block 1
  • Fb is the downward force of block 1 on block 2
  • Fc is the downward gravitational force of the earth on block 1
 Student 1's reasoning was:

We know that Fa is equal to Fb because they are a Newton's 3rd law force pair.  Because the blocks are at rest when the spring is maximally compressed, they are in equilibrium and thus forces on each block are balanced.  The forces acting on the top block are the force of block 2 upward (here called Fa), the force of the spring upward, and weight downward (here called Fc).  Setting up = down, we see that Fc has to be greater than Fa.  Final ranking: Fc > (Fa = Fb).

Which parts of this reasoning are right?  Which are wrong?

Correct: Fa and Fb are in fact a Newton's 3rd law force pair.  The force of block 2 on block 1 is equal to the force of block 1 on block 2.

Incorrect: The blocks are not in equilibrium at the instant of maximum compression.  Equilibrium means no acceleration, so no change in speed.  Here, the blocks may be at rest momentarily, but their speed is always changing!  For the blocks to be in equilibrium, they must be at rest and not changing their speed from zero.  

Incorrect: The force of the spring does not act on block 1!  All forces (in physics 1) other than gravity require contact.  The spring is not touching block 1, and so does not apply a force to block 1.

So how to solve, then?

Draw a correct free body diagram for the top block.  This has the weight downward (called Fc), and the upward contact force of block 2 on block 1 (called Fa).  The top block has an upward acceleration: when an object speeds up, its acceleration is in the direction of motion.  This block is starting to speed up and move upward, so its acceleration is also upward.*  Acceleration is in the direction of the unbalanced force - so the forces on the top block are unbalanced upward, meaning the upward Fa is greater than the downward Fc.  This, paired with the Newton's 3rd law pair, leads to the correct answer (Fa = Fb) > Fc.


*What about right before the block came to rest?  When an object slows down, its acceleration is opposite the direction of motion.  Just before the position of maximum compression, the block was slowing down and moving downwards... also giving an upward acceleration!




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