Worked Exercise — The Gorilla Pulls a Crate

Problem

A Gorilla grabs a crate of $m = 70\;\text{kg}$ with a rope that starts at the crate and points downward at $30^\circ$ below the horizontal (the rope passes between the gorilla’s hands higher up). It pulls with a force of $|\vec{F}_g| = 600\;\text{N}$. The coefficient of static friction between the crate and the floor is $\mu = 1$. Does the crate slide?

Solution

Step 1 — List of the forces. Four forces act on the crate: the weight $\vec{P}$, the floor’s normal reaction $\vec{R}$, friction $\vec{F}_a$ and the gorilla’s force $\vec{F}_g$.

Step 2 — Resolving into components. With the $x$ axis horizontal along the direction of pull and the $y$ axis vertical upward:

$\vec{P} = (0,\ -70\cdot 9{,}81) = (0,\ -686{,}7) \qquad \vec{R} = (0,\ R)$

$\vec{F}_g = (F_g\cos 30^\circ,\ -F_g\sin 30^\circ) = (519{,}6,\ -300) \qquad \vec{F}_a = (-F_a,\ 0)$

where friction opposes the impending motion to the right.

Step 3 — Equilibrium equations. Imposing $\sum\vec{F} = \vec{0}$ separately on the two axes:

$\begin{aligned} x: &\quad 519{,}6 - F_a = 0 \;\Rightarrow\; F_a = 519{,}6\;\text{N} \\ y: &\quad -686{,}7 + R - 300 = 0 \;\Rightarrow\; R = 986{,}7\;\text{N} \end{aligned}$

Step 4 — Comparison with the maximum friction. The maximum available static friction force is $F_a^\text{max} = \mu R = 1\cdot 986{,}7 = 986{,}7\;\text{N}$. The friction required for equilibrium is $519{,}6\;\text{N}$, and since

$\ev{F_a^\text{richiesta} = 519{,}6\;\text{N} < 986{,}7\;\text{N} = F_a^\text{max}}$

the crate does not slide.

Teaching note. By pulling downward, the gorilla increases the normal reaction $R$ and hence also the maximum available friction. If instead it pulled upward at the same angle, $R$ would decrease and it would be easier to make the crate slide.

Links

Topics: Statica ed equilibrio Concepts: Equilibrio statico · Componenti di un vettore · Forza peso · Forza normale · Attrito statico Skills: Scomposizione vettoriale Methods: Scomposizione in componenti cartesiane Objects: Blocco