AcademyEquilibrium and Materials
Academy
Solving Rigid-Body Equilibrium
Level 1 - Physics topic page in Equilibrium and Materials.
Principle
Rigid-body equilibrium is solved by matching independent balances to unknown reactions.
Notation
\(R_x,R_y\)
pin or hinge reaction components
\(\mathrm{N}\)
\(T\)
tension
\(\mathrm{N}\)
\(W\)
weight
\(\mathrm{N}\)
\(\tau_z\)
torque component about the chosen pivot axis
\(\mathrm{N\,m}\)
\(f_s,N\)
static friction and normal reaction
\(\mathrm{N}\)
\(\mu_s\)
static friction coefficient
none
Method
Derivation 1: Keep only the external forces
Free body
\[\text{external forces only}\]
Weight acts at the center of gravity; contacts act at contacts.
Unknown reactions
\[\text{solve for }R_x,\ R_y,\ T\ \text{or contact forces}\]
Derivation 2: Choose a pivot that removes the most unknowns
The beam sketch is set up so torque about the hinge removes the two hinge reactions from the first equation.
Zero torque through the pivot
\[\tau_{\mathrm{pivot}}(R_x)=\tau_{\mathrm{pivot}}(R_y)=0\]
Torque equation
\[\sum\tau_z=0\]
Derivation 3: Recover the remaining reactions from force balance
Once one unknown is found from moments, the horizontal and vertical balances determine the support reactions.
Horizontal balance
\[\sum F_x=0\]
Vertical balance
\[\sum F_y=0\]
Derivation 4: Check the contact model
Static friction does not automatically equal its maximum value; it takes whatever value is needed up to the limit.
Friction bound
\[|f_s|\le\mu_s N\]
Rules
These are the compact results from the derivations above.
Horizontal balance
\[\sum F_x=0\]
Vertical balance
\[\sum F_y=0\]
Torque balance
\[\sum\tau_z=0\]
Static-friction limit
\[|f_s|\le \mu_s N\]
Examples
Question
A uniform
\[4\,\mathrm{m}\]
beam of weight \[100\,\mathrm{N}\]
is hinged at one end and held horizontal by a vertical cable at the far end. Find cable tension.Answer
Torque about the hinge gives
\[T(4)-100(2)=0\]
so \[T=50\,\mathrm{N}\]
Checks
- Do not include internal forces.
- A pin can exert force but no torque.
- Friction takes the needed value up to its maximum.
- Negative contact force means contact is lost.