If 54 J of work are needed to stretch a spring from 15 cm to 21 cm and 90 J are needed to stretch it from 21 cm to 27 cm, what is the natural length of the spring?
To find the natural length of the spring, we can assume that the spring follows Hooke's Law, which states that the force required to stretch or compress a spring is directly proportional to the displacement. The formula for the potential energy stored in a spring is given by:
Potential Energy (PE) = (1/2)kx^2
where PE is the potential energy, k is the spring constant, and x is the displacement.
According to the problem, 54 J of work is needed to stretch the spring from 15 cm to 21 cm. We can calculate the displacement as:
Displacement 1 (x1) = 21 cm - 15 cm = 6 cm = 0.06 m
Using the formula, we have:
Potential Energy 1 (PE1) = (1/2)kx1^2 = 54 J
Similarly, 90 J of work is needed to stretch the spring from 21 cm to 27 cm. Calculating the displacement, we have:
Displacement 2 (x2) = 27 cm - 21 cm = 6 cm = 0.06 m
Using the formula again, we have:
Potential Energy 2 (PE2) = (1/2)kx2^2 = 90 J
Since the spring constant (k) is the same for both cases, we can set up the following equation:
PE1 = PE2
(1/2)kx1^2 = (1/2)kx2^2
Canceling out 1/2 and k, we get:
x1^2 = x2^2
Substituting the known values:
0.06^2 = 0.06^2
0.0036 = 0.0036
Since this equation is true, it means that the displacements are equal. This indicates that the natural length of the spring is the distance from the unstretched position (0 displacement) to the point where the first displacement occurs. Therefore, the natural length of the spring is 15 cm.
To find the natural length of the spring, we need to understand the relationship between the work done and the displacement of the spring.
The work done on a spring is given by the formula:
Work = (1/2)kx^2
where k is the spring constant and x is the displacement of the spring from its natural length.
We are given that 54 J of work are needed to stretch the spring from 15 cm to 21 cm, and 90 J are needed to stretch it from 21 cm to 27 cm.
Let's solve for k:
For the first displacement:
54 = (1/2)k(21-15)^2
54 = (1/2)k(6)^2
54 = (1/2)k(36)
k = 2*(54/36)
k = 3 J/cm^2
For the second displacement:
90 = (1/2)k(27-21)^2
90 = (1/2)k(6)^2
90 = (1/2)k(36)
k = 2*(90/36)
k = 5 J/cm^2
Now, we can use the spring constant to find the natural length of the spring.
For the natural length displacement:
0 = (1/2)k(x^2)
0 = (1/2)(3)(x^2)
0 = (3/2)x^2
Since the displacement cannot be negative, we take x as positive. Thus, x = 0.
Therefore, the natural length of the spring is 0 cm.