Some Examples of Static Equilibrium — Question 6

Question 6

Separated door hinges supply a horizontal force couple

A door of weight $300\,\mathrm N$ has its center $0.40\,\mathrm m$ horizontally from its vertical hinge line. Two hinges lie $1.50\,\mathrm m$ apart vertically. Model each hinge as a force connection with no individual couple; the door is static.

1. Find the horizontal reaction components in the plane of the door.

2. Can equilibrium alone determine the vertical force carried by each hinge?

Question 6 - Solution

Separated door hinges supply a horizontal force couple

(a) Derivation and calculation

Let $H_u,H_l$ denote horizontal hinge forces, positive away from the hinge line. The weight produces moment $$Wb=300(.40)=120\ \mathrm{N\,m}.$$ The hinge couple must oppose it: $$H_l(1.50)=120\quad\Longrightarrow\quad\boxed{H_l=80\ \mathrm N}.$$ Horizontal balance gives $\boxed{H_u=-80\ \mathrm N}$.

(b) Result and interpretation

The vertical reactions satisfy only $$\boxed{V_u+V_l=300\ \mathrm N}.$$ Any pair $V_u=q$, $V_l=300-q$ satisfies rigid-body balance. Because both act on the same vertical hinge line, their split does not alter the relevant moment. Hinge stiffness or fit information is required to determine $q$.

Physical check

The upper horizontal force points toward the hinge line and the lower one away from it, giving the required counterclockwise couple.