Hi could someone please help me answer this question:
It requires 0.05 joule of work to stretch a spring from a length of 8 cm to 9 cm and another 0.10 joule to stretch it from 9 cm to 10 cm. Determine the spring constant and find the natural length of the spring.
Thank you
To solve this problem, we need to use Hooke's Law, which states that the force required to stretch or compress a spring is directly proportional to the displacement of the spring from its natural length.
First, let's find the spring constant (k). The formula for work done on a spring is given by:
Work = (1/2) * k * (x^2)
Where:
- Work is the energy required to stretch the spring (given as 0.05 J and 0.10 J).
- k is the spring constant (unknown).
- x is the displacement of the spring (given as 1 cm).
For the first case, when the spring is stretched from 8 cm to 9 cm, the displacement (x) is 1 cm and the work (W) is 0.05 J. Plugging these values into the formula, we have:
0.05 J = (1/2) * k * (1 cm)^2
Simplifying, we get:
0.05 J = (1/2) * k
0.10 J = k
Now, for the second case, when the spring is stretched from 9 cm to 10 cm, the displacement (x) is again 1 cm, but the work (W) is given as 0.10 J. Plugging these values into the formula, we have:
0.10 J = (1/2) * k * (1 cm)^2
Simplifying, we get:
0.10 J = (1/2) * k
0.20 J = k
Since we have two different equations for k, we can equate them to find the spring constant:
0.10 J = 0.20 J
k = 0.20 J
Now that we know the spring constant (k), we can find the natural length of the spring (L). The formula for the natural length is given by:
L = x + (W/k)
Where:
- L is the natural length of the spring (unknown).
- x is the displacement of the spring (given as 1 cm).
- W is the total work done on the spring (0.05 J + 0.10 J).
- k is the spring constant (0.20 J).
Plugging in the values, we have:
L = (1 cm) + [(0.05 J + 0.10 J) / (0.20 J)]
L = 1 cm + (0.15 J / 0.20 J)
L = 1 cm + 0.75 cm
L = 1.75 cm
Therefore, the spring constant is 0.20 J and the natural length of the spring is 1.75 cm.