Help with proportionality and literal equations

[jxj]

Homework Statement

Its a series of problems essentially basically asking questions about solving proportionality
. For example
"Hooke's Law of a spring can be described by the equation F = -kx, where F is the force exerted by a spring, K us the spring constant, and X is the amount of distance a spring has been stretched.

A. Determine how much the force exerted by spring changes if K is tripled. B. Determine force exerted if X is decreased by a factor of 5. C. Determine the force exerted by a spring if the spring constant is decreased by a factor of three and X is quadrupled. D. Use the method of literal equations to isolate the spring constant and determine the dimensions in units of the spring constant.

Homework Equations

F= -kx

The Attempt at a Solution

I tried to take the approach of it increasing as the k or x increased, but I'm honestly lost lol. Please help

 

[lewando]

Why don't you take these one-at-a-time. Start with question A. If F₁ = -kx_{specific-nonzero-x} and F₂ = -3kx_{specific-nonzero-x} what is the ratio F₂/F₁?

 

[symbolipoint]

I'm not sure why you need the negative sign.

Just as generally, if some y is proportional to x, DIRECTLY or JOINTLY proportional, then for some constant k, you can say
y=kx.

Remember, k is a constant, and therefore, k=y/x.
If you know or expect the relationship is correct, then if you know one pair of x and y, then you can calculate or compute k. Once you know k value, you can find either x for any y value; or find y for any x value.

 

[jxj]

Why don't you take these one-at-a-time. Start with question A. If F₁ = -kx_{specific-nonzero-x} and F₂ = -3kx_{specific-nonzero-x} what is the ratio F₂/F₁?

wouldn’t that mean the force you have exerted, F, is also tripled?

 

[jxj]

I'm not sure why you need the negative sign.

Just as generally, if some y is proportional to x, DIRECTLY or JOINTLY proportional, then for some constant k, you can say
y=kx.

Remember, k is a constant, and therefore, k=y/x.
If you know or expect the relationship is correct, then if you know one pair of x and y, then you can calculate or compute k. Once you know k value, you can find either x for any y value; or find y for any x value.

thanks, but I am still a little confused on the proportionality between the two. I also put s negative because that's how hookes law is set haha

 

[jxj]

Why don't you take these one-at-a-time. Start with question A. If F₁ = -kx_{specific-nonzero-x} and F₂ = -3kx_{specific-nonzero-x} what is the ratio F₂/F₁?

you would essentially have -3kx/-kx, right?

 

[symbolipoint]

thanks, but I am still a little confused on the proportionality between the two. I also put s negative because that's how hookes law is set haha

The language of proportionality is about like "direct variation" and "joint variation". A simple example is like a line represented as y=x. Let x be each of several different values. Find y. Same value as x. Plot the line of points (x, y). What is the proportionality constant? y=kx ?
This is k=1. You could say, y=1*x. That is a simple, very simple example.

Another simple example:\
You may have these points: (1, 3), (2,6), (3, 9), and (4, 12).
You may choose to plot the points and draw the line.
Notice, the line intersects with (0,0).
Notice the slope for y/x is 3.
How you may represent this as an equation can be y=3x.

What can you say in written English language for this?
y is directly proportional to x. x is 5 when y is 15.

If you were given that, you could take as y=kx, and find the value for k.

y/x=k

k=y/x

k=15/5

k=3.

 

[haruspex]

not sure why you need the negative sign.

Because it is a good idea to take the same directions positive for all forces, accelerations, velocities and displacements in a given line. For a spring, the displacement and force are in opposite directions.

 

[lewando]

wouldn’t that mean the force you have exerted, F, is also tripled?

you would essentially have -3kx/-kx, right?

Yes and yes. So if you are clear with that, go ahead with the others.

 

[jxj]

The language of proportionality is about like "direct variation" and "joint variation". A simple example is like a line represented as y=x. Let x be each of several different values. Find y. Same value as x. Plot the line of points (x, y). What is the proportionality constant? y=kx ?
This is k=1. You could say, y=1*x. That is a simple, very simple example.

Another simple example:\
You may have these points: (1, 3), (2,6), (3, 9), and (4, 12).
You may choose to plot the points and draw the line.
Notice, the line intersects with (0,0).
Notice the slope for y/x is 3.
How you may represent this as an equation can be y=3x.

What can you say in written English language for this?
y is directly proportional to x. x is 5 when y is 15.

If you were given that, you could take as y=kx, and find the value for k.

y/x=k

k=y/x

k=15/5

k=3.

thanks. seeing it shown as (x,y) made it somewhat easier to understand

 

[jxj]

thanks. seeing it shown as (x,y) made it somewhat easier to understand

Looking at B-C Seeing as how it uses X now, would it still be directly proportional to F or now jointly? I would have said X being decreased by a factor of five would result in F being decreased being by 5, though i’m unsure if that’s correct.

also thanks for the explanation

 

[lewando]

Looking at B ... I would have said X being decreased by a factor of five would result in F being decreased being by 5, though i’m unsure if that’s correct.

Have you tried the method of post #2? If so, what is the source of your uncertainty?

 

[jxj]

Have you tried the method of post #2? If so, what is the source of your uncertainty?

Yes, I tried the process as I used for A. and now I am wondering won't F decrease by a factor of 5, if x is decreased by a factor of 5?

 

[lewando]

You are correct in your wonderment. But you should be pretty confident in your result. Whatever it is that is causing you to have any doubt is what I wanted to explore.

 

[jxj]

You are correct in your wonderment. But you should be pretty confident in your result. Whatever it is that is causing you to have any doubt is what I wanted to explore.

thanks! I think that for C, if K is decreased by a factor of three and X is quadrupled, F would also decrease and quadruple? or does two changes change the proportionality?

and for D. How do suppose I use the method of Literal Equations to isolate the spring constant and determine the units?

thanks for all your help!

 

[lewando]

I think that for C, if K is decreased by a factor of three and X is quadrupled, F would also decrease and quadruple?

Yes. Sounds contradictory but it is not. For fun (and educational purposes) this time, show your thoughts, step-by-step, à la post #2 method.

for D. How do suppose I use the method of Literal Equations to isolate the spring constant and determine the units?

I suppose the first step can start with you stating your understanding of what is meant by:
1) "method of Literal Equations to isolate the spring constant"
and
2) "and determine the units [of the spring constant]"

 

[jxj]

Yes. Sounds contradictory but it is not. For fun (and educational purposes) this time, show your thoughts, step-by-step, à la post #2 method.I suppose the first step can start with you stating your understanding of what is meant by:
1) "method of Literal Equations to isolate the spring constant"
and
2) "and determine the units [of the spring constant]"

Yes. Sounds contradictory but it is not. For fun (and educational purposes) this time, show your thoughts, step-by-step, à la post #2 method.I suppose the first step can start with you stating your understanding of what is meant by:
1) "method of Literal Equations to isolate the spring constant"
and
2) "and determine the units [of the spring constant]"

Lol I got it finally. it took some time to figure out directly and inversely proportionality.

 

[symbolipoint]

Lol I got it finally. it took some time to figure out directly and inversely proportionality.

Like said, the typical academic language is usually very easy to use. This is another way of handling constant or uniform rates examples or exercises.