Question

data for single spring two - spring arrangement
some students attach a single spring to a clamp and let the spring hang vertically. objects of different mass are attached to the free end of the spring and allowed to hang at rest. the students measure the distance (delta y) the spring stretches from its equilibrium length for each object. the students produce the graph of (delta y) as a function of the weight (f_{g}) of the objects shown in the figure, and the slope of the best - fit line to the data is determined to be (s_{1}). next, the students take a second spring that is identical to the first and arrange the two springs as shown in the two - spring arrangement next to the graph. once again, objects of different mass are attached to the two - spring arrangement, (delta y) is measured, and the data is plotted on another graph showing (delta y) as a function of (f_{g}). which of the following best describes the slope of the best - fit line to the data collected for the two - spring arrangement and gives correct supporting reasoning for the claim?
a it is less than (s_{1}). the sum of the forces exerted by each spring is the weight of the supported object which effectively increases the spring constant of the system.

Explanation:

Step1: Recall Hooke's Law

Hooke's Law is \(F = k\Delta y\), where \(F\) is the force applied, \(k\) is the spring constant, and \(\Delta y\) is the displacement. Rearranging for \(\Delta y\), we get \(\Delta y=\frac{F}{k}\). For a single spring, the slope \(S_{1}=\frac{1}{k_{1}}\).

Step2: Analyze two - spring arrangement (parallel)

When two identical springs (\(k_{1} = k_{2}=k\)) are in parallel, the equivalent spring constant \(k_{eq}=k_{1}+k_{2}=2k\). Then, for the two - spring arrangement, \(\Delta y=\frac{F}{k_{eq}}=\frac{F}{2k}\). The slope \(S_{2}=\frac{1}{k_{eq}}=\frac{1}{2k}\).
Since \(S_{1}=\frac{1}{k}\) and \(S_{2}=\frac{1}{2k}\), \(S_{2}

Answer:

A. It is less than \(S_{1}\). The sum of the forces exerted by each spring is the weight of the supported object which effectively increases the spring constant of the system.