# When are stairs NOT an inclined plane?

• Student800
In summary, a staircase with a length of 10 meters and a height of 5 meters has a mechanical advantage of 2.2.
Student800

## Homework Statement

What is the mechanical advantage of a staircase that is 10 meters long (horizontally) and 5 meters high (vertically)?

MA=Length/Height

## The Attempt at a Solution

MA=2.2

I'm certain that this is the answer my teacher is looking for, but it does not make sense to me. It seems to me that stairs are not always the same as an inclined plane, but that this may depend on the configuration of the steps and how they are climbed.

For example, what if the stairs have 2 equally spaced steps (each 2.5, meters high) or 4 equally spaced 1.25 meter high steps. I don't see how those configurations provide any mechanical advantage -- I have to climb each high step and then walk over the flat distance between steps. There is no reduction in the force I have to use to life my body.

In the more typical configuration, if I move my body horizontally over the flat part of the steps and vertically at the edge of the steps, how do the stairs provide any mechanical advantage? Again, there is no reduction in the force I have to use to lift my body. Furthermore, even if I were to move my body at a constant slope upward equal to the average slope of the stairs (as if I were a block sliding on a flat inclined plane), isn't it really me that preserves that slope? In other words, couldn't I climb a vertical ladder and essentially zigzag body back and forth as I climb so that my body center of mass is always moving up at any given slope? By my teacher's formula, the ladder has an MA=1.

I can only imagine that I'm missing something important here because my teacher said he's been using this same problem set since before I was born. Can someone please help me figure out where my reasoning has gone wrong?

Student800 said:
In the more typical configuration, if I move my body horizontally over the flat part of the steps and vertically at the edge of the steps, how do the stairs provide any mechanical advantage?
But that is not how one actually walks up steps. The legs bend and straighten in such a way that the mass centre moves in a reasonably straight line.

Yes, I see how walking up certain stairs in a certain way may provide for your center of mass to make a smooth path up the stairs. On a ladder I can make my center of mass approximate any path you might like despite what the formula might give for the mechanical advantage of a ladder.

I guess I was just stuck on:
1) the how important it is for the stair-walker to use their legs to make their center of mass track the incline path in order for the formula to work, and
2) how the force in the direction of the incline is only a component of the force your feet have to apply to move up the steps while for a traditional inclined plane analysis, the force in the direction of the incline is the only force applied to the weight.

Thanks again!

## 1. What is an inclined plane?

An inclined plane is a flat surface that is at an angle to the ground. It is often used to move objects from a lower level to a higher level with less force.

## 2. How are stairs different from an inclined plane?

Stairs are a type of inclined plane, but they have a series of steps that make them easier to climb. They also have a steeper angle compared to a typical inclined plane.

## 3. When are stairs not considered an inclined plane?

Stairs are not considered an inclined plane when they are curved or spiral, as they do not have a constant angle of incline. Additionally, stairs that have a landing or change in direction are not considered a continuous inclined plane.

## 4. What are some examples of stairs that are not inclined planes?

Spiral staircases, helical stairs, and grand staircases with landings or turns are all examples of stairs that are not considered inclined planes. These types of stairs have varying angles of incline or changes in direction, making them different from a traditional inclined plane.

## 5. Can stairs be classified as an inclined plane in certain situations?

Yes, stairs can be considered an inclined plane if the angle of incline remains constant and there are no changes in direction or landings. For example, a straight set of stairs with no turns or landings can be classified as an inclined plane.

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