Worked exercise — Ladder leaning against a smooth wall

Problem

A ladder of mass $m = 20\;\text{kg}$ and length $L = 3\;\text{m}$ leans against a smooth vertical wall (no friction). The ladder makes an angle of $60^\circ$ with the floor. The coefficient of static friction between ladder and floor is $\mu = 1$. Does the ladder slip?

Solution

Step 1 — Forces on the ladder.

• weight $\vv P = (0,\, -196{,}2)\;\text{N}$, applied at the ladder’s centre;
• floor reaction $\vv R_1 = (0,\, R_1)$, vertical, applied at the foot;
• friction at the floor $\vv F_a = (F_a,\, 0)$, horizontal, applied at the foot;
• wall reaction $\vv R_2 = (-R_2,\, 0)$, horizontal (the wall is smooth, no friction), applied at the top end.

Step 2 — Translational equilibrium $\sum\vv F = \vv 0$:

$\begin{aligned} x:&\quad F_a - R_2 = 0 \;\Rightarrow\; F_a = R_2 \\ y:&\quad R_1 - 196{,}2 = 0 \;\Rightarrow\; R_1 = 196{,}2\;\text{N} \end{aligned}$

Step 3 — Rotational equilibrium (clever pole at the foot). Choosing the pole at the foot of the ladder, $\vv R_1$ and $\vv F_a$ have zero lever arm and vanish. What remains is the weight (horizontal arm $\tfrac{L}{2}\cos 60^\circ = 0{,}75\;\text{m}$) and the wall reaction (applied at the top, height $L\sin 60^\circ \approx 2{,}60\;\text{m}$):

$\sum M = -P\cdot 0{,}75 + R_2\cdot 2{,}60 = 0 \;\Rightarrow\; R_2 = \frac{196{,}2\cdot 0{,}75}{2{,}60} \approx 56{,}6\;\text{N}$

Step 4 — Comparison with maximum friction. From $F_a = R_2 \approx 56{,}6\;\text{N}$, while the available static friction is

$F_a^{\text{max}} = \mu R_1 = 1\cdot 196{,}2 = 196{,}2\;\text{N}$

Since $56{,}6\;\text{N} < 196{,}2\;\text{N}$, the required friction is amply available:

$\ev{F_a \approx 56{,}6\;\text{N} < F_a^{\text{max}} = 196{,}2\;\text{N}\;\Rightarrow\; \text{the ladder does not slip}}$

Linked atoms

Argomenti: Statics and equilibrium Concetti: Static equilibrium · Moment of a force · Static friction · Normal force · Weight force Competenze: Moment balance · Choice of pole · Vector decomposition Metodi: Decomposition into Cartesian components Oggetti: Rigid body · Rod