If 27 J of work are needed to stretch a spring from 18 cm to 24 cm and 45 J are needed to stretch it from 24 cm to 30 cm, what is the natural length of the spring?
To find the natural length of the spring, we need to understand the relationship between work and the displacement of the spring.
The work done on a spring is given by the formula:
W = (1/2)kx²
Where W is the work done, k is the spring constant, and x is the displacement of the spring from its natural length.
Let's use the given information to set up two equations and solve for the spring constant, k.
Equation 1:
Given that 27 J of work are needed to stretch the spring from 18 cm to 24 cm, we can write:
27 = (1/2)k(0.06²)
Simplifying the equation:
27 = (1/2)k(0.0036)
Divide both sides by 0.0036:
27 ÷ 0.0036 = k
7500 = k
Equation 2:
Given that 45 J of work are needed to stretch the spring from 24 cm to 30 cm, we can write:
45 = (1/2)k(0.06²)
Simplifying the equation:
45 = (1/2)k(0.006)
Divide both sides by 0.006:
45 ÷ 0.006 = k
7500 = k
Since we obtained the same value of k from both equations, we can conclude that the spring constant, k, is equal to 7500 N/m.
Now, we can find the natural length of the spring by setting the displacement, x, to zero in the equation:
W = (1/2)kx²
This gives us:
0 = (1/2)k(0)
0 = 0
Since the work done and displacement are both zero, the natural length is when the spring is at equilibrium and not stretched or compressed. So, the natural length of the spring is the length of the spring when it is not stretched at all.
Therefore, to find the natural length, we look at the original length before any stretching occurred. From the given information, we know that the spring stretched from 18 cm to 24 cm and then further stretched from 24 cm to 30 cm. Thus, the original length or the natural length of the spring is 18 cm.