4.2.1.P6A

The normal force that occurs when two objects touch has what seems on the surface to be a strange property: it adjusts itself depending on other circumstances. If you put a block on a horizontal table the table will exert a normal force on the block equal and opposite to the block's weight (since the block doesn't move). But if you put a second block on top of the first, the normal force the table exerts on the block that is resting on it will adjust itself to equal the sum of the weights of the two blocks. See, for example, the problem The farmer and the donkey. In that problem, the blocks are resting on a spring scale, and it is easy to see that when you add more weight, the spring will compress more and therefore exert more force. Could that be what happens on a table too? But the deformation of a table when you put something on it seems very small. Surely a deformation so small you can't see it can't matter? Let's see if a simple model of a set of many many tiny springs (atom-atom interactions) can work to explain normal force. We'll do this in a sequence of problems. For this first problem, let's just see how many springs connected together add up.

Since it's easier to think about how springs respond to being pulled rather than to being compressed, we'll talk about it using our "pull-pull" picture discussed in the reading Springs where springs being pulled from opposite directions (but not too much!) stretch according to Hooke's law: T = kΔL. The result is the same for compressions as for extension except for the direction (signs) of the tension forces. 

We'll take as our basic element a single spring (model of an atom-atom interaction) with rest length ℓ0 and spring constant, k, as shown in the figure at the right. If it is pulled from opposite directions by a tension force, T, it will stretch an amount Δℓ that satisfies the equation  T = kΔℓ where its stretch length = ℓ0 + Δℓ.

First let's consider the effect of linking atoms together end to end into a long chain. This kind of connection is referred to as connected in series.

A. Let's start by considering two identical springs each having spring constant, k, linked together and pulled from opposite ends by equal tension forces T. We imagine that they are connected together by molecules that are short and that don't stretch significantly compared to the springs themselves.

A1. Consider this combination as a single "effective" spring. (Imagine it's contained in a box and you can't see that it actually is made up of two separate springs.) How much would it stretch in when it is pulled from the two ends by a tension force T ?

1. ℓ₀ .
2. Δℓ.
3. 2Δℓ.
4. Δℓ/2.
5. Something else. (What?)

A2. If we define the effective spring constant of the combination by the equation T = k_eff ΔL, where ΔL is the total amount the combination stretches, how does k_eff compare to k?

1. k_eff = k 
2. k_eff = 2k 
3. k_eff = k/2
4. Something else. (What?)

A3. Now suppose that we have attached not two springs end to end, but N of them.  Write an equation that expresses the effective spring constant of the combination with the spring constant of the original spring, k, and the number of springs, N.

A4. How does the combination of many springs connected in series compare as a spring to the individual components?

1. It is a softer spring (easier to stretch)
2. It is a stronger spring (harder to stretch)
3. It is the same as the components (equally hard to stretch)
4. There is not enough information to decide. 

Next let's consider the effect of linking springs to the same connecting and ending point. This kind of connection is referred to as connected in parallel.   B. Consider the same two identical springs each having spring constant, k, but this time with both linked at each end to the same point and pulled from opposite ends by equal tension forces T.

B1. Consider this combination as a single "effective" spring. (Imagine it's contained in a box and you can't see that it actually is made up of two separate springs.) How much would it stretch when it is pulled from the two ends by a tension force T ? 

1. ℓ₀ .
2. Δℓ.
3. 2Δℓ.
4. Δℓ/2.
5. Something else. (What?)

B2. If we define the effective spring constant of the combination by the equation T = k_eff ΔL, where ΔL is the total amount the combination stretches, how does k_eff compare to k?

1. k_eff = k 
2. k_eff = 2k 
3. k_eff = k/2
4. Something else. (What?)

B3. Now suppose that we have attached not two springs, but N of them.  Write an equation that expresses the effective spring constant of the combination with the spring constant of the original spring, k, and the number of springs, N.

B4. How does the combination of many springs connected in parallel compare as a spring to the individual components?

1. It is a softer spring (easier to stretch)
2. It is a stronger spring (harder to stretch)
3. It is the same as the components (equally hard to stretch)
4. There is not enough information to decide. 

Joe Redish 9/13/15