created at April 12, 2021

# Lesson 5 - Equilibrium and reactions

## Equilibrium

Equilibrium is a core concept of statics. Equilibrium of the body means that body is in the rest state relative to surrounding bodies. In other words, the body is static in some desired coordinate system. For instance, your house is in rest state (in equilibrium state) relative to the earth. The house isn't moving anywhere despite a set of forces is applied to him: gravity, background friction force, pressure of the fallen tree above the roof:

• Ft - fallen tree pressure force
• Fg - gravity force
• Ff - friction force
• Fn - reaction force

Another example is a couch inside a train's car. You are loading it with your weight's force while sitting on it. On the other hand, reaction force denies the coach from falling through the floor, and friction force denies it from sliding. So, the coach is in equilibrium state relative to car, despite the car itself is moving on a railroad.

• F - passengers weight
• Fg - weight of the couch
• Ff - friction force between the couch and the floor
• Fn - floor's reaction force

The most essential thing is:

The body can be in equilibrium state only when resulting force and torque vectors are zeros. No matter how complex the initial system of loads is. If the resulting force is zero - the body will remain static (no translations). This is not enough since the body still can rotate with no translations. In order to ensure no rotations the resulting torque vector must also be zero. So, zero resulting force and zero resulting torque are two necessary conditions for equilibrium state. Let's write down it as follows:

$\vec{R}=0;\quad \vec{M_{O}}=0;\quad (1)$

where Mo is a torque relative to some desired point O, which can be selected randomly.

This is the core concept of Newton's first law.

In other words, when the force F is applied to the body there should be another force F', which has the same magnitude but opposite direction in order to compensate F and ensure equilibrium of the body. For instance, we have placed the glass of water on the table. The gravity force of the glass tries to move it along the force application line. So, as a result the glass should fall down through the table. However, the glass still stays in rest state. That's because the table is acting on the glass with the same force (as gravity), which has the same magnitude and opposite direction.

Now it is easy to derive equations of statics from the equilibrium condition (1). We will do that in next lesson, but now let's talk about connections.

## Connections

The body is assumed to be FREE when it can move in any ways without any restrictions. For instance, a soccer ball in flight is a free body. Another example is a spaceship, flying on the orbit.

Connections are the other bodies, which somehow restrict the possible motions of desired body. When we apply a force to the body, connection begins to act on that body with another force, which tries to prevent body from moving. This "another" force is called reaction force. Reaction force acts in opposite to that direction, where the initial force tries to move the body.

Reaction forces were already mentioned in previous lessons for many times (Reaction of the table, reaction of the railroad etc.).

Despite reaction forces appear as a result of interaction between the desired body and it's connections, you should treat them just as regular force in such sense that they are real and also contribute equations of statics. By the way, reaction forces are usually unknown and are obtained by solving equations of statics.

Most connections met in practice can be reduced to several common types regarding on how they restrict the motion of the body. This is very useful in order to simplify solving of problems. Let's find out what they are.

## Flat surface

Absolutely flat surface prevents body from moving through it. Surface is considered with no friction. When the body touches the surface, the surface begins to react, thus creating an obstacle for further body moving. The reaction force of the surface is directed along the surface normal. If the surface is not ideal and has friction, then reaction force can also have tangent component.

Thread is connecting body with some point A, preventing that body move away from this point, starting from a certain distance. The reaction force is directed along the thread's line towards point A. Thread can only be stretched. When body is moving towards point A, no reaction occurs. "Thread" is just a name of the connection type. In practice it can be anything that behaves the way thread does: rope, cable, chain etc.

By the way, the ball on the figure above is not in equilibrium state. The gravity force Fg and rope reaction force N are applied to it, however, they are not coincident, thus they can't compensate each other. The resulting force R is not zero and it forces the ball to move towards the house.

## Cylindrical joint

This is a connection type when the body allowed to freely rotate around some axis. It can also slide along the axis, but it can't move away from axis. As example let's take a look at the engine's crankshaft (fig. 1.5.6). The crankshaft is supported by sliding bearings, so they provide a "cylindrical joint" type connection to him. Piston rods are impacting the crankshaft with their loads trying to move it, but reaction forces arising in sliding bearings prevents crankshaft from moving, allowing rotation only.

For cylindrical joint the vector of reaction force lies in the plane, which is orthogonal to rotation axis:

Another good example of cylindrical joint is a radial ball bearing. However, the most commonly used bearings are angle contact bearings. They can take a load component in axial direction as well. Then the reaction force may have its component along the axis.

## Spherical joint

Spherical joint ensures the body rotation around some point O and prevents any translations relative to that point. The good example is a camera joint (fig. 1.5.8). Reaction force vector goes through the point O and it can have any direction.

## Weightless rod

Weight of such a rod is negligible in comparison with load it carries. As example we can mention again a piston rod. It connects a piston with a crankshaft. Reaction force of such a rod is directed along its axis as in the case with thread. However, in opposite to thread the rod can take both tension and compression load, so the actual direction of rod's reaction force is generally unknown.

reaction forces N and N' are equal but opposite directed. Force N impacts the piston from rod's side and force N' impacts the crankshaft from rod's side.

## Clamp

As example let's consider a metal tube immured into the wall. Such connection prevents any movement, both translation and rotation. The reaction may include all possible components - 3 unknown force components and 3 unknown torque components. In the case of 2D problem (in plane problem) we obtain two unknown force components and one unknown torque.

And finally its worth to clarify some things about the equilibrium. As was told before (for the sake of simplicity), equilibrium is a rest state relative to surrounding bodies. If more exact, equilibrium means that the body can either be in rest state or move straightforward with constant velocity. In other words, there is no difference between being static and moving straightforward with constant velocity. Why so? Because in both cases resulting forces and torques are zero.

Everything in the world tries to preserve its current state, and it actually does until some force impacts it. It is called Inertia. If the body was moving under a set of applied loads - it will continue moving with its current velocity preserving current direction, after all loads were released.

If it's still not clear, consider the example with house again. It remains static (equilibrium state) relative to the earth. However, the earth itself is moving around the sun with constant velocity (about 29 km/s). And the solar system itself is also moving somehow in the universe. But it doesn't change the essence - the house still stays in equilibrium state relative to earth as well as the couch still stays in equilibrium state relative to train's car etc.