Static equilibrium Knowledge-Based Study

Study notes

1. What is Static Equilibrium?

   An extended rigid body is in static equilibrium when it remains completely at rest, exhibiting neither linear acceleration nor angular acceleration relative to an inertial reference frame.

   The Two Equilibrium Conditions

   For point particles, equilibrium requires only that the vector sum of all external forces is zero. However, for extended bodies where forces can be applied at different locations, two independent vector conditions must be satisfied simultaneously:

   1. Translational Equilibrium (First Condition): The net external force acting on the body must be zero:
      $\sum \vec{F}_{\text{ext}} = 0$
      In two dimensions, this resolves into two scalar equations along orthogonal axes: $\sum F_x = 0$ and $\sum F_y = 0$.
   2. Rotational Equilibrium (Second Condition): The net external torque acting on the body about any chosen reference axis must be zero:
      $\sum \vec{\tau}_{\text{ext}} = 0$
      For planar problems, choosing a perpendicular rotation axis yields: $\sum \tau = 0$.

   If the first condition is satisfied but the second is not, the object experiences angular acceleration while its center of mass remains at rest or moves at a constant velocity.

   Calculating Torque

   Torque measures the rotational effectiveness of a force about an axis. For a force $F$ applied at a position vector $\vec{r}$ from the chosen pivot point, the magnitude of torque is defined as:

   $\tau = r F \sin\theta = F d_{\perp}$

   • $r$: Distance from the pivot point to the point of force application.
   • $\theta$: Angle between the position vector $\vec{r}$ and the force vector $\vec{F}$.
   • $d_{\perp} = r \sin\theta$: Lever arm (moment arm), which is the perpendicular distance from the pivot line to the line of action of the force.

   Sign Convention and Pivot Freedom

   By convention, counterclockwise (CCW) torques are designated as positive, and clockwise (CW) torques are negative:

   $\sum \tau = \sum \tau_{\text{CCW}} - \sum \tau_{\text{CW}} = 0$

   Because the body is in translational equilibrium ($\sum \vec{F} = 0$), the net torque is independent of the reference point. Therefore, you can place the imaginary pivot point at any convenient location. Choosing a pivot point through which one or more unknown forces act eliminates their torques (since $r = 0$), greatly simplifying the system of equations.

   Center of Gravity

   The gravitational force acts on every individual mass element of an extended body. In a uniform gravitational field, the net gravitational effect is equivalent to a single resultant force, the total weight $W = mg$, acting directly at the body's center of gravity (CG).

   For uniform geometric objects (such as uniform rods, disks, or rectangular plates), the center of gravity coincides exactly with the geometric center (centroid).

   Tipping versus Slipping

   When external horizontal forces act on an object supported on a surface, the normal force does not necessarily act at the center of the base. To maintain rotational equilibrium, the line of action of the normal force shifts toward the edge about which the object tends to tip:

   • Stable Contact: The line of action of the resultant normal force falls within the boundary of the support base.
   • Impending Tipping: The normal force shifts all the way to the pivot edge. If the disturbing torque exceeds the restoring torque exerted by weight about that edge, the object begins to rotate and tip over.
   • Impending Slipping: The horizontal force reaches the maximum static friction value ($f_{s,\text{max}} = \mu_s N$) before the normal force reaches the tipping edge.

   Step-by-Step Methodology

   Static equilibrium problems involving beams, struts, ladders, and cables can be solved reliably using the following structured approach:

   1. Isolate the Object and Draw an Extended Free-Body Diagram (FBD): Draw the body showing the precise point of application, direction, and magnitude of each external force (including normal, friction, hinge/pin, and tension forces).
   2. Set Up a Coordinate System: Decompose all forces into convenient horizontal and vertical components (or parallel and perpendicular to the body).
   3. Apply Translational Equilibrium: Write equations for $\sum F_x = 0$ and $\sum F_y = 0$.
   4. Select an Optimal Pivot: Choose a pivot point that lies on the line of action of the largest number of unknown forces (such as a hinge or support with unknown reaction forces).
   5. Apply Rotational Equilibrium: Calculate each torque about this pivot with its proper sign, and set $\sum \tau = 0$.
   6. Solve the System of Equations: Solve for the remaining unknowns and check that your results are physically reasonable (e.g., tensions must be positive).