To find the length of the spring when no weights are attached, we can use the information provided about the lengths of the springs when weights are applied.
Let's denote:
- \( L_0 \) = the length of the spring when no weights are attached.
- \( W_1 = 2 \) pounds, which causes the length \( L_1 = 12 \) inches.
- \( W_2 = 5 \) pounds, which causes the length \( L_2 = 18 \) inches.
To find the rate of change of the length of the spring with respect to the weight, we can calculate the change in length per pound of weight.
-
The change in length when going from a weight of 2 pounds to a weight of 5 pounds is: \[ \Delta L = L_2 - L_1 = 18 , \text{inches} - 12 , \text{inches} = 6 , \text{inches} \] The change in weight is: \[ \Delta W = W_2 - W_1 = 5 , \text{pounds} - 2 , \text{pounds} = 3 , \text{pounds} \]
-
Now, we calculate the rate of change of length per pound: \[ \text{Rate of change} = \frac{\Delta L}{\Delta W} = \frac{6 , \text{inches}}{3 , \text{pounds}} = 2 , \text{inches per pound} \]
-
Now, we can express the length of the spring with respect to the weight applied. If we assume the relationship between weight and length is linear, we can use the following equation where \( L = L_0 + k \cdot W \), where \( k \) is the rate of change: \[ L = L_0 + 2W \]
-
Plugging in the values we know, we can solve for \( L_0 \) using either the first or second spring:
- For \( W = 2 \): \[ 12 = L_0 + 2 \cdot 2 \implies 12 = L_0 + 4 \implies L_0 = 12 - 4 = 8 , \text{inches} \]
- For \( W = 5 \): \[ 18 = L_0 + 2 \cdot 5 \implies 18 = L_0 + 10 \implies L_0 = 18 - 10 = 8 , \text{inches} \]
Both calculations confirm that the length of the spring when no weights are attached is:
8 inches
2 answers