## Equilibrium

For an object to be in mechanical equilibrium, the net external force and the net external torque acting on the object have to be zero.

Ftot = 0,  τtot = 0.

The total torque can be nonzero, even though Ftot = 0.  Consider the situation shown in the figure on the right.  The square is initially at rest in the x-y plane.  The z-axis points out of the page.  The total force on the square is zero.  The CM of the square will stay at rest.  The total torque about the CM is not zero.  The square will have angular acceleration pointing into the page.

No net external force implies that the center of mass of the object is at rest or moving with constant velocity.  No net external torque implies that the object either does not rotate or that it rotates with constant angular velocity.  For an object in mechanical equilibrium there exists some inertial reference frame in which the object's center of mass is at rest.  If in this frame the object also does not rotate, it is in static mechanical equilibrium.

#### Problem:

A 1500 kg automobile has a wheelbase (the distance between the axles) of 3 m.  The center of mass of the automobile is on the centerline at a point 1.2 m behind the front axle.  Find the force exerted by the ground on each wheel.

Solution:
The car is in static equilibrium, Ftot = 0, τtot = 0.
The force of gravity Mg acts on the center of mass of the object.  It produces no torque about the CM.
Ftot = F1 + F2 - Mg = 0.
τtot = F2(1.8 m) - F1(1.2 m) = 0.
F1 = 1.5 F2.
2.5 F2 = Mg = 14700 N, F2 = 5880 N.  F1 = 8820 N.
The force exerted by the ground on each rear wheel is F2/2 = 2940 N and the force exerted by the ground on each front wheel is F1/2 = 4410 N.

When an object is placed in a uniform gravitational field, then the force of gravity produces no torque about the center of mass of the object.  We can always represent the gravitational force as an arrow pointing straight towards or away from the center of mass of the object.  For this reason, the center of mass is also called the center of gravity of the object.

### Equilibrium does not guaranty stability.

#### Example:

A ball and a bowl
The ball is in stable equilibrium at the bottom of a bowl.

The ball is in unstable equilibrium at the top of a bowl.

Objects are in stable mechanical equilibrium if their potential energy increases when they are slightly disturbed.

An object in static equilibrium has static translational stability, if, when displaced from its equilibrium position by a small amount, a restoring force accelerates it back towards the equilibrium position.

An object in static equilibrium has static rotational stability, if, when rotated away from its equilibrium orientation through a small angle, a restoring torque results in angular acceleration back towards the equilibrium orientation.

An object has static rotational stability if its center of gravity is located above its base of support.  A minimum of 3 non-collinear contact points form a base.

When an object with static stability is disturbed too much, the limit of static stability can be reached.  The limit of static rotational stability is reached when the object's center of gravity is no longer located over its base of support.

A tricycle has a base with 3 contact points.  It has has static rotational stability.

A bicycle has only 2 contact points.  It is statically unstable and tips over easily.