4.2.1.P6

The muscles in animals are complex structures consisting bundles of long fibers that contract in response to chemical signals. When contractile fibers are lengthened and when they are bundled, how the resulting pull changes depends on the physics of how connected springs add their forces. In order to understand how this works, let's consider a toy model of a muscle: many springs connected in different ways.

Since we are modeling with springs and metal springs do not contract on their own, let's think about how they respond to being pulled rather than to pulling. The result is the same except for the direction (signs) of the tension forces. We could construct a mechanical model that would pull, but this model is more straightforward to think about.

We'll take as our basic element a single spring (model of a fiber) 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 fibers together end to end into a longer fiber. (In a biological muscle, multiple cells combine actually creating single long cells with multiple nuclei.) 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 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?)

3.  2Δℓ. Each of the two springs is pulled from both sides by a tension force T, so each stretches by an amount Δℓ. The total stretch is therefore twice that.

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?)

3.  k_eff = k/2. Since the stretch is the combined strings is ΔL = 2Δℓ, our equation can be written 

T = k_eff 2Δℓ. Since T = kΔℓ, we can put this in for T to get

kΔℓ =  2k_eff Δℓ.

Solving for k_eff  gives the answer #3.

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.

Using similar reasoning, we get k_eff = k/N.

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

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

1. It is a softer spring, since the same pull produces a longer stretch. 

Next let's consider the effect of linking fibers 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?)

4.  Δℓ/2. Each of the two springs shares the tension force T, so each only gets pulled by T/2 and stretches by an amount Δℓ/2. 

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?)

2.  k_eff = 2k. Since the stretch is the combined strings is ΔL = Δℓ/2, our equation can be written 

T = k_eff Δℓ/2. Since T = kΔℓ, we can put this in for T to get

kΔℓ =  k_eff Δℓ/2.

Solving for k_eff  gives the answer #2.

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.

Using similar reasoning, we get k_eff = Nk.

B4. How does the combination of many springs connected in parallel, thought of as a single spring, compare 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. 

2. It is a stronger spring, since the same pull produces a shorter stretch. 

Joe Redish  9/6/15