Equipment: Slinky spring, plastic ruler, weight scale, stand with clamp and boss head.

Use the value of the acceleration due to the gravity $g=9.8~{\rm m/s}^2$.

Part A. Lengths of single coils (5.0 points)

A1  ^0.40 Count the total number of coils (turns) $N$ of the Slinky spring.

A2  ^0.20 Determine the mass $M$ of the entire Slinky spring.

If the Slinky spring is suspended by one end, the extension of its turns in the upper and lower parts will differ. Number the spring coils from top to bottom and denote the coil number by $n$. Let $L_n$ denote the distance between the beginning of the $n$-th turn and the end of the $(n+1)$-th turn (i.e., the length of two turns — the $n$-th one and the next one).

A3  ^2.00 Secure the spring in a stand clamp by its top turn. Measure the dependance $L_n$ on $n$ over the widest possible range. Perform at least 20 measurements.

A4  ^0.60 Plot the graph of $L_n$ versus $n$.

Let us proceed to the theoretical description of the Slinky. Let $m_0$ be the mass of a single turn, and $k$ be the stiffness coefficient of a single turn. As before, we will number the spring turns from top to bottom. Let $l_n$ denote the distance between the beginning and the end of the $n$-th turn. 

Consider the system of bodies consisting of the $n$-th turn and all the others located below it.

A5  ^0.40 Sketch the picture and indicate the external forces acting on this system of bodies.

Note that in the undeformed state, the spring turns are pressed tightly against each other. Therefore, the deformation $\Delta l_n$ of the turn, which appears in the expression for the elastic force $F_{\text{el}}=k\Delta l_n$, will be considered equal to its length $l_n$ (i.e. $F_{\text{el}}=kl_n$).

A6  ^0.40 Using the equilibrium condition, express $l_n$ in terms of $m_0$, $g$, $n$, $N$, and $k$.

In question A3, measurements were taken of the length $L_n$ of two turns — the $n$-th and the $(n+1)$-th. Therefore, $L_n=l_n + l_{n+1}$.

A7  ^0.40 Express $L_n$ in terms of $m_0$, $g$, $n$, $N$, and $k$.

A8  ^0.60 Using the graph plotted in question A4, as well as the measurement results in questions A1 and A2, determine the stiffness coefficient $k$ of a single turn.

Part B. Mass of the Slinky spring (5.0 points)

If you place the Slinky spring on a scale and vertically lift its upper end to a height $H$, the scale readings will decrease.

First, turn on the scale, then place the Slinky spring on it. Secure the top turn of the spring in the stand clamp. By adjusting the position of the stand clamp, one can change the value of $H$.

B1  ^2.20 Measure the dependence of the scale readings $m$ on $H$. Perform at least 11 measurements.

Let us investigate the dependence of $m$ on $H$ theoretically. As in the first part of the problem, let us number the turns from top to bottom. Again let $l_n$ denote the deformation of the $n$-th turn. Note that during each of the experiments, only a part of the spring's turns, located at the top, is deformed. Let $X$ be the number of spring turns that are deformed. The remaining $N-X$ turns simply lie on the scale.

In this case, the extension of the entire spring (which is the value $H$) can be determined as:
$$H=\sum_{n=1}^X l_n.$$

Mathematical Hint. A number sequence of the form $a$, $a+d$, $\ldots$, $a+(n-1)d$, $\ldots$ is called an arithmetic progression. Each number in such a sequence, starting from the second, is obtained from the previous one by adding a constant number. The general form of the $n$-th term of the sequence is $a_n=a_1+(n-1)d$.

The following formula holds for the sum $S_X$ of the first $X$ terms of an arithmetic progression:
$$S_X=\sum_{n=1}^X a_n=\frac{a_1+a_X}{2}\cdot X.$$

B2  ^0.60 Using the result of question A6, express $H$ in terms of $X$, $m_0$, $g$, and $k$. Note that the number of deformed turns is $X$, not $N$.

Consider the $N-X$ turns lying on the scale as a system of bodies.

B3  ^0.40 Sketch the picture and indicate the external forces acting on this system of bodies. What is the elastic force acting on the system?

B4  ^0.40 Express the scale readings $m$ in terms of $M$, $X$, $m_0$ and $g$.

Hint. When solving the next question, you may assume that the number of deformed turns is $X \gg 1$. In this case, in particular, $X(X+1)\simeq X^2$.

B5  ^0.40 Using the results of questions B2 and B4, propose a linearization of the dependence $m(H)$.

B6  ^0.60 Plot the linearized graph of the dependence $m$ versus $H$.

B7  ^0.40 Using the plotted graph, determine the stiffness coefficient $k$ of a single turn.