integral work problem?
So if 2 J (joules) of work is needed to stretch a spring from its natural length of 30cm to 42cm, how much work is needed to stretch the spring from 35cm to 40cm?
Please can anyone help with this? I did this so far but its not right:
2=k(0.12), k=50/3
integral from (.05 to .1) (50/3)x dx --> (25/3)x^2 from .05 to .1
Update:The answer the book provides is 25/24 ~ 1.04 J
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Your problem is the calculation of k... Work is NOT kx, as you did at first, but rather the integral of kx (as you did in your second step). I'm not sure why you didn't use the integral for both.
2 = ∫ kxdx (from 0 to 0.12)
2 = (1/2)kx² (from 0 to 0.12)
2 = (1/2)k(0.12)² - (1/2)k(0)
2 = 0.0072k
k = 2/0.0072 = 277.777777....
Use this value of k now, and you should get the right answer :)
The work needed to stretch a spring from its natural length by x cm is
W = 1/2 k (x/100)^2 where we divided by 100 so that the work is in joules and k is in N / m
Therefore, 2 = 1/2 k (12/100)^2 = 1/2 k (.12)^2 = 1/2 k (.0144)
So k = 4 / .0144 = 277.778 N / m
Now the work needed to stretch the spring from 35 cm to 40 cm is
1/2 k ( (0.4-0.3)^2 - (0.35 - 0.30)^2 ) = 1/2 (277.778) ( 0.01 - 0.0005) = 1.319 J