Worked exercise — two crates with a spring and pulley

Problem

Crate 1 ($3\;\text{kg}$) is attached to a spring ($K = 100\;\text{N/m}$, $\ell_0 = 1\;\text{m}$) anchored to the wall, and to a rope that passes over a pulley and holds crate 2 ($4\;\text{kg}$) vertically. The spring is stretched: $\ell = 2\;\text{m}$. Find the acceleration of the crates.

Solution

Step 1 — Interaction diagram. Circling crate 1, we count 4 outgoing squiggles: the weight (Earth), the spring’s elastic force (wall), the rope’s tension (crate 2) and the floor’s constraint reaction. Four forces therefore act on crate 1.

Step 2 — Free body of crate 1 (horizontal motion). The spring is stretched by $\Delta\ell = \ell - \ell_0 = 2 - 1 = 1\;\text{m}$, so it pulls towards the wall with $F_m = K\,\Delta\ell = 100 \cdot 1 = 100\;\text{N}$ directed along $-x$. The weight is $P_1 = m_1 g = 3 \cdot 9{,}81 \approx 29{,}4\;\text{N}$. We write $\vv{P}_1 + \vv{R}_1 + \vv{F}_m + \vv{t}_1 = m_1\,\vv{a}$:

$\begin{pmatrix} 0 \\ -29{,}4 \end{pmatrix} + \begin{pmatrix} 0 \\ R_1 \end{pmatrix} + \begin{pmatrix} -100 \\ 0 \end{pmatrix} + \begin{pmatrix} t \\ 0 \end{pmatrix} = 3\begin{pmatrix} a \\ 0 \end{pmatrix}$

$y$ axis: $R_1 = 29{,}4\;\text{N}$. $x$ axis: $-100 + t = 3a \quad \text{(I)}$

Step 3 — Free body of crate 2 (vertical motion). Crate 2 hangs from the rope; if crate 1 accelerates towards $+x$ (towards the pulley), crate 2 accelerates downward with the same magnitude $a$:

$-m_2 g + t = -m_2 a \quad\Rightarrow\quad -39{,}2 + t = -4a \quad \text{(II)}$

Solving the system. From (II) we get $t = 39{,}2 - 4a$ and substitute into (I):

$-100 + 39{,}2 - 4a = 3a \quad\Rightarrow\quad a = -\frac{60{,}8}{7}$

$\ev{a \approx -8{,}7\;\text{m/s}^2}$

The negative sign indicates that crate 1 accelerates in the direction opposite to the one assumed: the spring, very taut, overcomes the tension and pulls the system back. The tension is obtained from (II), $t = m_2(g - a)$:

$t = 4\,\bigl(9{,}8 - (-8{,}7)\bigr) = 4 \cdot 18{,}5$

$\ev{t \approx 74\;\text{N}}$

Links

Topics: Dinamica Concepts: Seconda legge di Newton · Tensione · Forza elastica e legge di Hooke · Forza normale Skills: Diagramma di interazione · Diagramma di corpo libero Methods: Metodo del diagramma di corpo libero Objects: Molla · Puleggia · Blocco · Fune ideale