a force of 1250 pounds compresses a spring 5 inches from its natural length. find the work done in compressing the spring 8 additional inches.
f=kd
1250=k5
k=250
then what do i do. i have to do this the calculus way, not the physics way. by using integrals
the answer choices are:
a.3250
b.21,125
c.18000
d.2000
e. none of these
are you in grade 12? do you wear ties and hats? is your middle name caleb? (im serious! i think i know you!)
and i am not sure about the question... sorry!
To find the work done in compressing the spring 8 additional inches, you can use calculus and the concept of work as the integral of force with respect to distance.
First, let's find the expression for the force exerted by the spring as a function of the compression. Since the force is directly proportional to the compression, we can write:
F(x) = kx
where F(x) is the force applied by the spring, k is the spring constant (which you already found to be 250), and x is the compression.
Now, we want to calculate the work done in compressing the spring an additional 8 inches from its original compression of 5 inches. We can set up the integral as follows:
Work = ∫[from 5 to 13] F(x) dx
Substituting F(x) = kx, we have:
Work = ∫[from 5 to 13] (kx) dx
Using the value of k as 250, the integral becomes:
Work = ∫[from 5 to 13] (250x) dx
To find the antiderivative of 250x, we add 1 to the exponent and divide by the new exponent:
Work = 250 * ∫[from 5 to 13] x dx
Integrating x with respect to x gives us:
Work = 250 * [(x^2)/2] [from 5 to 13]
Plugging in the limits of integration:
Work = 250 * [(13^2)/2 - (5^2)/2]
Calculating the expression inside the brackets:
Work = 250 * [(169/2) - (25/2)]
Work = 250 * [144/2]
Work = 250 * 72
Work = 18,000
Therefore, the work done in compressing the spring 8 additional inches is 18,000. Hence, the answer is option c: 18,000.
As for your personal questions, I am an AI bot designed to provide explanations and answers. I do not wear ties or hats, and my middle name is not Caleb.