Mechanical balance

Mechanical balance of the material point

All solid bodies have a shape and some dimensions, but sometimes they can be treated in the simplest way, using the theoretical model of material point. In this case the shape and dimensions of the body are neglected and it is considered that all its mass is concentrated in a single point.

A type body material point is in mechanical equilibrium if:

is at rest
- or
it moves in a uniform straight line

relative to an inertial frame of reference.

The necessary and sufficient condition for the mechanical equilibrium of the material point is that the resultant of all forces is 0.

Mechanical equilibrium of the rigid solid

In many cases, a body cannot be reduced to a simple material point. The easiest to treat is the body rigid solid, which does not deform, but maintains its shape and dimensions unchanged during its mechanical evolution.

To locate a rigid solid body, we must specify the position of a representative point with respect to a reference system, but also its orientation with respect to a set of reference directions.

A rigid solid body can perform two types of motion:

translational movement: the body maintains its fixed orientation during movement;
rotational movement: the body permanently changes its orientation around a fixed axis;

or a combination thereof (movement composed of translation and rotation).

Example

Mechanical system made up of components that are rigid solid bodies:

Translational balance

A rigid solid body is in translational equilibrium if:

is at rest
- or
performs a uniform translational motion.

The necessary and sufficient condition for translational equilibrium is that the resultant of all forces is 0.

Example

A rigid solid body moves (by translation) on the inclined plane. Under certain conditions, its state is translational equilibrium:

Rotational balance

A rigid solid body is in rotational equilibrium if:

is at rest
- or
rotates uniformly about an axis.

Moment of force

Moment of force is the determining factor of a rotational movement.

Consider a crank arm on which a force acts $\vec{F}$ :

The effect is that this body will rotate. The further the force acts from the axis of rotation, the stronger the effect of rotation will be.

Moment of force is a physical vector quantity whose size is defined by:

$M = r \cdot F$

Unit of measure:

${⟨ M ⟩}_{SI} = 1 Nm$

By convention:

M>0 if the rotation will be counter-clockwise;
M<0 if the rotation will be clockwise.

Different phases of the rotational movement, in which the same rotational moment acts on the body:

The magnitude of the moment of force also depends essentially on the direction of the force relative to the direction of the crank arm. If between these directions the angle is 30° then the moment of the force is halved:

In general, the moment of force is determined by the formula:

$M = b \cdot F = r \cdot F \cdot \sin (α)$

The moment of the force is maximum when α=90° and becomes 0 when α=0.

The necessary and sufficient condition of rotational balance is that the resultant moment of all forces is 0:

$\vec{M_{R}} = \vec{M_{1}} + \vec{M_{2}} + ... = 0$

An example of application is leverage:

a) first-order leverage:

a variant of first-order leverage:

b) the second lever:

c) the lever of the 3rd orin:

Another example: A coupling is made between two mechanical shafts by means of 2 and 3 clamping screws, respectively:

Problems:

1. On a bicycle pedal having the length arm r=170mm a force of magnitude is applied F=300N, which keeps its direction unchanged during rotation. Determine the moment of force for different phases of rotation:

movement phase	power arm b	the moment of the force M

2. A car with mass m=1.6 tons climbs a 15% incline at constant speed. The wheels are of size R15 190 65: their diameter is 634mm, and the rims are fastened in 5 bolts evenly placed on a circle with a diameter of 110mm. We assume that the weight is distributed evenly on all wheels. What force does each screw apply? Displacement resistance forces are neglected.

wheel detail: