The distance a spring stretches is directly proportional to the force applied to the spring. A 4-lb weight stretches a spring 20 in. A 16-lb weight stretches a spring 26 in. Find the length of the spring with no weight attached.
6 inches
The length a spring stretches is directly proportional to the force applied. If a force of x pounds stretches a spring h inches, how much force is necessary to stretch the same spring m inches? (Assume x = 7, h = 4, and m = 15.)
With no weight the spring will be 18 in. Every 1-Lb stretches the spring 1/2 In. The spring moves 6 inches (from 20 to 26 inches) with an incremental 12 Lbs (4Lb to 16LB)
To solve this problem, we can use the concept of direct variation or direct proportionality. According to the problem statement, the distance a spring stretches is directly proportional to the force applied to the spring.
Let's denote the force applied to the spring as F, and the distance the spring stretches as d. We can set up a proportion to relate the force and distance as follows:
(F₁ / d₁) = (F₂ / d₂)
where F₁ and d₁ represent the force and distance for the first scenario, and F₂ and d₂ represent the force and distance for the second scenario.
In the first scenario, a 4-lb weight stretches the spring 20 inches. So, F₁ = 4 lb and d₁ = 20 in.
In the second scenario, a 16-lb weight stretches the spring 26 inches. So, F₂ = 16 lb and d₂ = 26 in.
Plugging in the given values into the proportion, we get:
(4 / 20) = (16 / 26)
Now, we can solve this proportion to find the constant of proportionality. Cross-multiplying, we get:
4 * 26 = 20 * 16
104 = 320
Simplifying, we find that the constant of proportionality is 0.325.
Now, to find the length of the spring with no weight attached, we can use the first scenario where no weight is attached. In this case, the force on the spring is 0 lb. Let's denote the length of the spring with no weight attached as d₀.
Using the constant of proportionality, we can set up the equation:
(0 / d₀) = (4 / 20)
Simplifying, we get:
0 = 0.325 * d₀
Dividing both sides by 0.325, we find:
d₀ = 0
Therefore, the length of the spring with no weight attached is 0 inches.