Basic question about basic concept: proportionality

In summary, the concept of being directly proportional means that there is a linear relationship between two variables and the value of one variable increases or decreases in direct proportion to the value of the other variable. However, when the relationship is described as being directly proportional to the square or square root of a variable, it indicates a non-linear relationship where the value of one variable increases or decreases at a faster rate compared to the other variable. This can be represented through algebraic equations, but finding these equations can be challenging when working with measured data.
  • #1
Ontophile
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This is not homework. I'm reading a physics for laypeople book and I'm trying to wrap my head around something that other people obviously have little or no trouble understanding.

I understand what it means when someone says that a body's acceleration is directly proportional to the magnitude of the impressed force. It means that if you apply one more unit of force, you'll get one more unit of acceleration out of it. Double the force applied, double the resulting acceleration; triple the force, triple the acceleration.

What I don't understand is what to imagine when someone says that something is directly proportional to the square of something, or directly proportional to the square root of something. Is being "directly proportional to the square root" of something the same as being "inversely proportional to the square" of something?

1) Suppose we lived in a universe in which a body's acceleration was directly proportional to the square of the impressed force. What would that mean? Would it mean that if I doubled the impressed force, I'd quadruple the resulting acceleration? Or would it mean that if I wanted to double the acceleration, I'd have to quadruple the impressed force?

2) Likewise, say a body's acceleration was directly proportional to the square root of the impressed force. What would that mean?

3) Suppose that tripling the impressed force yielded 6 times the acceleration, and that quadrupling the force yielded 8 times the acceleration. What sort of proportional relationship would that be?

4) Suppose that adding 1 unit of force yielded 1 unit of acceleration, and that adding 2 units of force yielded 1.5 units of acceleration, and that adding 3 units of force yielded 1.8333... units of acceleration, and that adding 4 units of force yielded 2.08333... units of accceleration. What sort of proportional relationship would that be?

5) Suppose that adding 1 unit of force yielded 1 unit of acceleration, and that adding 2 units of force yielded 1.5 units of acceleration, but that adding 3 units of force yielded 1.75 units of acceleration, and that adding 4 units of force yielded 1.875 units of acceleration. What sort of proportional relationship would that be?

Thanks in advance for your help...
 
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  • #2
'Directly proportional' is a verbal description for the algebraic equation:
y = ax,
in which y is said to be directly proportional to x.

Otoh, if you see the expression
y=ax+b,
you would say that y is proportional to x - i.e it gets bigger as x gets bigger but will not equal zero when x is zero. 'Directly' means there is no extra 'constant' term (b).

If you say that y is directly proportional to x2, that would be written as
y=ax2

or, if y is directly proportional to root x, then

y=a√x

Using the maths may bring on the pains but is nicely precise and you can tap it into your calculator if you know the numbers.

If you want to find the equation for some process by looking at some measured results, that is a bit harder because you, effectively, have to try to 'solve' an equation for which there is not always a simple solution or a simple expression to show the relationship. Scientists are always being presented with this sort ofd problem and sometimes there are horrific sums needed to produce a formula.
 

1. What is proportionality?

Proportionality is a mathematical concept that describes the relationship between two variables. It states that as one variable increases or decreases, the other variable increases or decreases in a consistent and predictable manner.

2. How is proportionality represented?

Proportionality is often represented through a mathematical equation, such as y = kx, where y and x are the two variables and k is a constant value. It can also be represented graphically with a straight line passing through the origin.

3. What is the difference between direct and inverse proportionality?

In direct proportionality, as one variable increases, the other variable increases by the same factor. In inverse proportionality, as one variable increases, the other variable decreases by the same factor. This is represented by y = kx for direct proportionality and y = k/x for inverse proportionality.

4. How is proportionality used in science?

Proportionality is used in science to describe and predict relationships between variables. It is a fundamental concept in many scientific fields, such as physics, chemistry, and biology. It allows scientists to make accurate predictions and understand the behavior of natural phenomena.

5. Can proportionality ever break down?

While proportionality is a useful concept, it can break down in certain scenarios. For example, if the relationship between two variables is not linear, proportionality cannot accurately describe it. Additionally, external factors or limitations may also affect the proportionality between variables.

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