Pho.rs: Coils of a weighty spring

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

1 \[N \in [75, 85]\]

0.40

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

1 \[M \in [180, 205]~\mathrm{g}\]

0.20

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.

1 Measurements $L_n$ от $n$	20 × 0.08
2 \[n_\mathrm{max}-n_\mathrm{min} \geq 60\]	0.40

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

1 The graph is plotted	0.60
2 Labels of axes are missing	-0.10
3 Poor scale	-0.10
4 Poor ticks	-0.10
5 There is no linear approximation	-0.10

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

1 Diagram is present	0.10
2 There are gravity and elastic forces in the diagram and they have correct direction	2 × 0.10
3 There are gravity and elastic forces in the diagram and they have correct application points	2 × 0.05

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

1 \[F_\mathrm{el}=m_0 (N-n+1) g\]or any other expression which contains qualitively correct dependence on $n$	0.30
2 \[l_n = (N+1-n)\frac{m_0 g}{k} \]	0.10

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

1 \[L_n = l_n + l_{n+1}\]	0.20
2 \[L_n = \frac{(2N-2n+1)m_0 g}{k}\]	0.20

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.

1 \[\mathrm{slope} \in [-0.76,-0.56]~\mathrm{mm}\]Can't be evaluated without units	0.30
2 \[k \in [60, 80]~\mathrm{N}/\mathrm{m}\]	0.30

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

1 Измерения $m$ от $H$	11 × 0.18
2 $H_\mathrm{max} \geq 25.0~\mathrm{cm}$	0.22

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 $N$.

1 \[H = \sum_{n=1}^{X} l_n\]	0.10
2 \[H=\frac{X(X+1)}{2} \frac{m_0 g}{k}\]	0.50

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?

1 A diagram is present	0.10
2 There are forces in the diagram and they have correct direction	2 × 0.05
3 There are gravity and normal reaction forces in the diagram and they have correct application ponits	2 × 0.05
4 Explicitly stated that $F_\mathrm{el}=0$	0.10

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

1 Equilibrium condition for a given system	0.10
2 \[m=M-m_0X\]	0.30

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

1 $H$ vs $(M-m)^2$ or $\sqrt{H}$ vs $m$

0.40

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

1 A linearized graph it plotted	0.60
2 Labels of axes are missing	-0.10
3 Poor scale	-0.10
4 Poor ticks	-0.10
5 There is no a linear approximation line	-0.10

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

1 The value of slope is determined Can't be evaluated without units	0.20
2 \[k = [60; 85]~\mathrm{N}/\mathrm{m}\]	0.20

Coils of a weighty spring

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