# Translational Equilibrium

Consider the cart below. The only force acting on the cart is the weight. On a horizontal road the cart is in equilibrium – the downward force due to gravity is exactly balanced by the upwards reaction on the wheels. The net vertical force on the cart is zero, so the cart does not move in the vertical direction. Likewise there is no net force in the horizontal direction, so the cart does not translate in that direction.

If the cart does not move horizontally or vertically, then it does

not move in any direction (because any movement can be resolved into horizontal and vertical components). This is a concept that we will use often – if the translation in two perpendicular directions is zero, then the\ translation in any other direction must be zero also, as any translation can be resolved into components in the two perpendicular directions.

Likewise, if the force in any two perpendicular directions is zero, then the force in any other direction must be zero, as any force can be resolved into components in the two perpendicular directions. Both displacement and force are vectors as they have a magnitude and a direction. All vectors can be resolved into components.

So our cart has translational force equilibrium. Expressed mathematically,

The cart is now placed on an inclined slope. Again the only force acting is that due to gravity. The wheels are free to rotate, so the reactions can only be directed perpendicular to the road surface.

The cart is no longer in equilibrium. To see this, resolve the load into components parallel and perpendicular to the road surface. The perpendicular component is balanced by the reaction, so we have translational equilibrium in that direction, but there is no reaction to balance the component of the load which acts parallel to the road surface. A lack of equilibrium is acceptable for a cart, which is intended to roll from place to place, but in a structure it is unacceptable.

Translational equilibrium occurs when every force acting on a structure (every load) has an equal and opposite resisting force (reaction)

F_{1} to F_{6} are** LOADS and REACTIONS** in any direction. For equilibrium, we require **ΣF=0**.

Many structures can be considered to be two dimensional – that is they can be drawn in the plane of a page. For these structures we can ensure that translational equilibrium is satisfied by choosing two perpendicular axes (call them x and y), and making sure that and ΣFx=0 and ΣFy=0.

In the more general case, and for more complex structures, we will have forces and reactions acting in all three directions. In these cases we need to satisfy,

**Required Reading (20 minutes):**- Read: Hibbeler, R.C., Statics and mechanics of Materials, Section 2.3.