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§4

Equilibrium of a body in the absence of the rotation.

Let a solid body, which has finite dimensions, to limit its motion for some reason in such a way, that it cannot rotate. For example, it can be a rod on an inclined plane, or a piston in a cylinder. If a body is in the equilibrium then the acceleration of its center of mass should be equal to zero. It is known from the dynamics that .

From this it follows that the sum of all external forces , acting to a body, is also equal to zero.

Hence, the condition of the equilibrium of a solid body in the absence of rotation in some inertial frame of reference should be the equality to zero of the sum of all external forces, acting to a body:

The condition of the equilibrium (4) is identical to the condition of the equilibrium (1) for the material point. It is not accidentally, because in the absence of rotation a body may be considered as a material point; this was already mentioned in the dynamics.

The vector equality (4) gives the possibility to write the condition of the equilibrium of a body in the form of three scalar equations . All said in § 3 about the coordinate axes, the equivalence of the equalities, the necessity, but not sufficiency of the conditions of the equilibrium remains valid here.

PROBLEM 2

The bar of mass m = 1 kg lays at the inclined plane with inclination angle . Let?s find the friction force between the bar and the inclined plane (see Figure 5).

SOLUTION

The force of gravity , the force of rest friction and the force of normal pressure act to the bar. For a moment we will not consider the question of the point of application of the force , because it is not significant for the solution of this problem. The condition of the equilibrium of the bar is: . It is not required to find the unknown force in this problem. Therefore we will direct x axis perpendicular to this force and will write the condition of the equilibrium of the bar in the projections to this axis: . From this we have: .

INDEX

§1. Introduction. §2. Force. Equivalence of forces. Resultant force. Addition of forces. Decomposition of force. §3. Equilibrium of a material point. §4. Equilibrium of a body in the absence of the rotation. §5. Equilibrium of a body with a fixed axis of the rotation in the planar case. Moment of the force. Test questions. Problems.
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