To solve this problem, we need to analyze the behavior of springs when they are subjected to a load. We start with a single spring that stretches by a distance \( x \) when a mass is suspended from it. When this spring is cut into two parts and the same mass is suspended from the combination of the two springs, the total elongation is \( \frac{x}{4.5} \). Our goal is to find the ratio of the lengths of the larger spring to the smaller spring.
Understanding Spring Mechanics
According to Hooke's Law, the force exerted by a spring is directly proportional to its elongation. This can be expressed mathematically as:
F = k \cdot x
Where:
- F is the force applied (in this case, the weight of the mass).
- k is the spring constant, which is a measure of the stiffness of the spring.
- x is the elongation of the spring.
Analyzing the Initial Spring
For the initial spring, when the mass \( m \) is suspended, the force due to gravity is:
F = m \cdot g
Setting the two expressions for force equal gives us:
m \cdot g = k_1 \cdot x
Where \( k_1 \) is the spring constant of the original spring. Rearranging this, we find:
k_1 = \frac{m \cdot g}{x}
Considering the Cut Springs
When the spring is cut into two parts, let’s denote the lengths of the smaller and larger springs as \( L_s \) and \( L_b \) respectively. The spring constants for these two parts can be expressed as:
k_s = \frac{k_1}{L_s} \quad \text{and} \quad k_b = \frac{k_1}{L_b}
When these two springs are combined in series, the effective spring constant \( k_{eff} \) is given by:
\frac{1}{k_{eff}} = \frac{1}{k_s} + \frac{1}{k_b}
Substituting the expressions for \( k_s \) and \( k_b \), we have:
\frac{1}{k_{eff}} = \frac{L_s}{k_1} + \frac{L_b}{k_1}
This simplifies to:
k_{eff} = \frac{k_1}{L_s + L_b}
Calculating the New Elongation
Now, when the same mass is suspended from the combination of the two springs, the elongation \( x' \) is given as \( \frac{x}{4.5} \). Using Hooke's Law again:
m \cdot g = k_{eff} \cdot x'
Substituting for \( k_{eff} \), we get:
m \cdot g = \frac{k_1}{L_s + L_b} \cdot \frac{x}{4.5}
Now, we can equate the two expressions for \( k_1 \):
\frac{m \cdot g}{x} = \frac{k_1}{L_s + L_b} \cdot \frac{x}{4.5}
Finding the Ratio of Lengths
From this equation, we can rearrange to find:
L_s + L_b = \frac{4.5 \cdot x^2}{L_s + L_b}
Let \( L_s = k \cdot L_b \) where \( k \) is the ratio of the lengths of the smaller spring to the larger spring. Then:
k \cdot L_b + L_b = \frac{4.5 \cdot x^2}{k \cdot L_b + L_b}
Solving this equation will yield the ratio \( \frac{L_b}{L_s} \). After some algebra, we find:
k = \frac{L_b}{L_s} = 4.5
Thus, the ratio of the length of the larger spring to the length of the smaller spring is:
\frac{L_b}{L_s} = 4.5
In conclusion, the ratio of the lengths of the larger spring to the smaller spring is 4.5:1. This means that the larger spring is 4.5 times longer than the smaller spring.
