a spring has a natural length of 20meters.
a) find the work required to stretch the spring from 20m to 30m
b) from 30m to 40m
c) how is the work from part a and part b related?
To find the work required to stretch a spring, we can use Hooke's Law, which states that the force exerted by a spring is directly proportional to the displacement from its natural length. The formula for the force exerted by a spring is given by:
F = -kx
Where:
F is the force applied to the spring,
k is the spring constant, which represents the stiffness of the spring,
x is the displacement from the natural length.
a) Work required to stretch the spring from 20m to 30m:
First, we need to find the change in displacement, ∆x = x_final - x_initial = 30m - 20m = 10m.
Since we are given only the natural length and not the spring constant, we need more information to directly calculate the work. However, if we assume that the spring is ideal and obeys Hooke's Law, we can find the work by integrating the force with respect to displacement.
The equation for work is:
W = ∫ F dx
In this case, the integral becomes:
W = ∫ (-kx) dx
W = -k ∫ x dx
W = -k * (x^2 / 2) + C
Since we are given that the spring has a natural length of 20m, we can set x = 0 when the spring is at its natural length. Therefore, the lower limit of integration is 0.
W = -k * (x^2 / 2) | 0 to 10
W = -k * [(10^2 / 2) - (0^2 / 2)]
W = -k * (100 / 2) = -50k
So, the work required to stretch the spring from 20m to 30m is -50k.
b) Work required to stretch the spring from 30m to 40m:
Using the same approach, we find the change in displacement, ∆x = x_final - x_initial = 40m - 30m = 10m.
Similarly, we integrate the force with respect to displacement:
W = ∫ (-kx) dx
W = -k * ∫ x dx
W = -k * (x^2 / 2) + C
Since we are given that the spring has a natural length of 20m, the lower limit of integration is 0.
W = -k * (x^2 / 2) | 0 to 10
W = -k * [(10^2 / 2) - (0^2 / 2)]
W = -k * (100 / 2) = -50k
Therefore, the work required to stretch the spring from 30m to 40m is also -50k.
c) The work from part a and part b is the same:
Both part a and part b involve stretching the spring by the same displacement of 10m. Since the work is determined by the change in displacement, which is the same in both cases, the work required to stretch the spring is also the same in both cases.