Work Problem - Springs

[Kaitlyn25]

I'm reviewing for a final, so I'm sorry if I have a few more questions than usual, just trying to clear up my understanding.

Problem:

A force of 50 N is required to hold a spring that has been stretched from its natural length of 12 cm to a length of 14 cm. How much work is done in stretching the spring from 14 cm to 20 cm? (Just set up the definite integral but do not integrate!)

I set it up like this:



However, my teacher had listed the upper limit as being .08, rather than .06. Is this because the total distance the spring has been stretched is from 12 cm to 20 cm? I guess I'm confused because it specifically says "how much work is done in stretching the spring from 14 cm to 20 cm?" Or does this type of problem automatically imply the total distance, even if it specifies a certain distance?

 

[Deleted member 4993]

I'm reviewing for a final, so I'm sorry if I have a few more questions than usual, just trying to clear up my understanding.

Problem:

A force of 50 N is required to hold a spring that has been stretched from its natural length of 12 cm to a length of 14 cm. How much work is done in stretching the spring from 14 cm to 20 cm? (Just set up the definite integral but do not integrate!)

I set it up like this:

View attachment 3252

However, my teacher had listed the upper limit as being .08, rather than .06. Is this because the total distance the spring has been stretched is from 12 cm to 20 cm?

Correct

I guess I'm confused because it specifically says "how much work is done in stretching the spring from 14 cm to 20 cm?" Or does this type of problem automatically imply the total distance, even if it specifies a certain distance?

Your teacher is correct.

The spring stretched from 0.02 m (14-12) to 0.08 m (20 - 12).

 

[HallsofIvy]

I'm not sure what distinction you are making between "total distance" and "certain distance"- and perhaps you aren't. If you look carefully at the integral you wrote, \(\displaystyle \int_{.02}^{.06} 2500xdx\), you will see that the lower limit is the "total distance" from the natural length while the upper bound is the distance from the 14 centimeter length, not from the natural length. You are mixing different kinds of distances. For both lower and upper limits you have to use the distance from the natural length.

 

[Kaitlyn25]

You're right, I am mixing the two types of distances. For the lower limit, I took the distance from the natural length of 12cm to 14cm, which is .02, but then for the upper limit, I took the distance from 14cm to 20cm when I should have taken the distance from the natural length to 20cm.

Thank you.