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kittykat219's avatar

Can you help me with this linear model question?

Asked by kittykat219 (136points) October 2nd, 2011

I’ve been assigned this question but I don’t really understand it. :/

There is a spring that is the length of 8cm when a 20 gram weight hangs at the bottom. Each additional gram stretches the spring another 0.15 cm. I’m supposed to write a linear model for this.

I want to say that 0.15 equals the slope and that X is 8 and Y is 20 but then it doesn’t work out. Do you have any ideas?

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14 Answers

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kittykat219's avatar

No it’s for my algebra class.

PhiNotPi's avatar

Here is what you do.

First, 0.15 is the slope because as x (the weight) increases by 1, y (the length) increases by 0.15. To find the equation for the line, we need to use the point (8, 20).

We set up an equation as follows
y = mx + b.
y = 0.15 x + b (The slope is 0.15)
20 = 0.15 * 8 + b (We know that when x=8, y=20)

Now solve for b.
20 = 1.2 + b
b = 18.8

You now know that b (the y-intercept) is 18.8. Since you now know both the slope and the y-intercept, you can write a linear equation.
Y = 0.15 X + 18.8

kittykat219's avatar

Thank you so much. :)
The only thing is that the next question asks me what will the length of the spring be when the weight equals 10 grams?
But when I substitute 10 for X in the equation, it doesn’t work..

PhiNotPi's avatar

@kittykat219 Why doesn’t it work, and how do you know that it doesn’t work?

kittykat219's avatar

Because when I put it into the equation the answer is 20.3 cm
But how is this possible because when a 20 gram hangs at the bottom of the spring the length is 8 cm..

Mariah's avatar

I’m thinking that x actually equals mass hanging and y equals spring length. This is because “each additional gram stretches the spring another 0.15 cm,” and it is the x value that is multiplied by .15 in the equation y = .15x + b. This implie that for each additional unit x a unit of .15x will be added to y, which is the relationship stated in the problem between mass and spring length. Try that.

kittykat219's avatar

So would the equation be Y=0.15X+5
Which would then make 6.5 the length of the spring when the 10 grams is hanging from it?

bobbinhood's avatar

@Mariah I think that’s what @PhiNotPi said, but then he(?) accidentally switched the numbers when he wrote them as a point and set up the equation.

@Mariah and @PhiNotPi Correct me if I’m wrong, but shouldn’t we get a directly proportional relationship between the stretch and the weight? So b should have to be zero? Physics isn’t my strong suit, but that seems to be what I remember.

PhiNotPi's avatar

Oops, I did accidentally put that the spring was 20 cm long when the weight was 8 grams. I did say that x was weight and y was length. It was just a simple mistake, but it completely destroys the answer.

It should be
8 = 0.15*20 + b
b = 5

Mariah's avatar

Yes, you have it right now.

Gravity will stretch the spring out a little bit before a weight is ever added, so it makes sense for it to have a nonzero value y even when x is zero.

PhiNotPi's avatar

Even when springs don’t have weight on them, they still have a length determined by the number and thickness of the coils, because it is physically impossible to fit the coils into no space.

bobbinhood's avatar

Thanks for clearing that up, guys. So, the stretch itself is zero when no force is applied, but the overall length obviously is not.

Mariah's avatar

@bobbinhood The stretch is zero when no force is applied but as the spring in question is hanging vertically, the force of gravity applies to the spring itself even when there is no additional weight added to the spring. Hang a spring vertically (preferably one that is not very stiff, to see the effect more clearly) and you will see that it stretches out a bit. Oftentimes, however, to make physics problems easier, the problem will specify a “massless” spring – obviously not a realistic concept but one that simplifies the math – in which case the stretch will be zero until additional weight is added. I suspect that might be what is going on in this problem, and the original 5cm that the equation specifies is simply the length of the unstretched spring. It doesn’t really matter either way in this context.

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