Static Equilibrium

The condition of Static Equilibrium depends on two conditions:

1. Translational Equilibrium: sum of all forces = 0 (that is, F_net = 0).

An object may be rotating, even rotating at a changing rate, but may be in translational equilibrium if the acceleration of the center of mass of the object is still zero.

2. Rotational Equilibrium: sum of all torques about a point on an object = 0; (net torque = 0).

An object may be accelerating in a linear fashion (along a straight line and or even turning at a constant rate; an object in rotational equilibrium will NOT be accelerating in a rotational sense (ie. the angular momentum of an object in rotational equilibrium will be constant).

Static Equilibrium is attained by satisfying BOTH of the above conditions at the same time.

Problem Types:

Objects in Translational Equilibrium:

Objects that are suspended (by contact with other objects or by tension forces) without acceleration are in translational equilibrium. These problems are solved either by resolving forces into components, or if there are 3 forces on the object, by adding the forces in a triangle and solving for the missing value.

Example #1: Find the tension in the horizontal string in the diagram below:

Solution:

tan 55^o = mg/F_T

F_T = (15kg)(9.8m/s²) / tan 55^o

F_T = 103 N

This solution can also be obtained using components of forces on the object. Two equations can be created from the sum of hoizontal forces = 0, and the sum of vertical forces = 0. From substitution (and a bit of labor), the final answer will be the same as above.

Objects in Rotational Equilibrium:

When an object has forces applied to it at some distance from the center of mass of the object, these forces cause a "turning force" on the object. The actual direction of the "turning force" or Torque, is perpendicular to a plane formed by the force on the object, and the displacement of this force from a chosen pivot point. We will consider Torque as a "turning force" in 2 dimensions.

t= F x d sin q

the sin q term defines the force as the PERPENDICULAR component of force applied at that point on the lever, or beam.

To find an unknown force on a beam, or lever arm in static equilibrium, set up the equation for the sum of torques applied by all forces on the beam = 0.

Establish the preferred pivot point (place at the most undesireable force location as it makes any torque due to that force equal to zero).

Establish the preferred coordinate axes (along the beam and perpendicular to it). Resolve all NON-perpendicular forces into components to find the parts causing torque on the beam.

Solve for the unknown force using the sum of torques = 0 equation.

Example #2: Find the force of the wall on the ladder in the diagram below:

Solution: Consider the pivot at the base of the ladder.

St = 0
0 = (F_wcos30)d - (F_gcos60)d/2
**since beam is uniform, the c.o.m. is at d/2
F_w = 34 N