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## Applied solid dynamics / D.G. Gorman, W. Kennedy - Details - Trove

No notes for slide. The notion of force expresses a mechanical action on an object. The forces have a well identified physical origin, for example the gravitational force weight , the electrical force on a charged particle , the contact or tension force on a cable, or the elastic force of a spring. Like many fundamental notions of physics, force is difficult to define in itself, but it is approached through experimental examples as well as through the mathematical relationships it has with other quantities.

Figure 1. Equilibria of opposite forces, a in a spring dynamometer, b in a hot air balloon. The sum of the forces acting on a resting object must cancel each other out.

A force can be measured by the elongation of a spring from its rest position Figure 1. Experimentally, it is verified that this elongation is proportional to the force by successively adding several identical weights.

## Applied Solid Dynamics

Once calibrated, this spring force gauge can be used to measure different forces. This is none other than the result of the atmospheric pressure forces acting all around the envelope: due to the pressure decrease with altitude, the pressure is higher at the bottom of the envelope than at the top, which translates into a net upward force. It is thanks to this balance that air plots, or water plots within a basin, do not fall to the ground under the influence of gravity.

When the air is heated, its density and therefore the mass of a given volume decreases, while the pressure remains unchanged because it is controlled by the weight of the surrounding air. The balance is then disrupted leading to a vertical acceleration of the balloon. In the atmosphere, an air mass heated locally by solar radiation will also tend to rise: this is the principle of convection.

Figure 2.

The equilibrium shape is such that at each wire intersection the vector sum of the forces is equal to zero. Each beam element is represented by a wire, and its mass simulated by a weight. The tension force on a yarn is necessarily aligned with the yarn. In the actual inverted configuration, the corresponding force will then be a compression force aligned along the beam, which guarantees its mechanical strength. More generally, the balance of forces must be expressed as vectors , which is the basis for calculating structures in architecture, see Figure 2.

For each material point, for example the intersection node of the wires, the vector sum of the forces must cancel each other out at equilibrium, as shown in Figure 3a. An extended object, for example a solid , is described in physics as a set of material points held together by internal forces. These forces are to be distinguished from external forces such as weight or forces of contact with other objects. The sum of the internal forces is cancelled out by the principle of action and reaction , so that equilibrium requires the cancellation of the sum of the external forces.

But the equilibrium condition of an extended object also requires the cancellation of the total moment of the forces, to avoid its rotation. The moment of a force with respect to an axis is defined as the product of the force, projected perpendicular to the axis, by the distance to the axis. The classic example is the lever shown in Figure 3b. It is usual to consider the moments in relation to the axis of the lever, because the moment of the reaction force R of the ground cancels itself out.

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However, the same result could be obtained by calculating the moment with respect to any mathematical axis, adding the moment of the reaction R , which is an equal vector and opposite to the sum of the two forces F1 and F2. In the absence of force, an object moves at a uniform speed, this is the principle of inertia first stated by Galileo This principle was not very intuitive at the time, because in everyday life any movement tends to stop in the absence of effort.

This deceleration negative acceleration is now attributed to friction forces, which are opposed to speed. In fact, we consider the limit of a very short time interval.

For a sufficiently high horizontal velocity, the curvature of the Earth must then be taken into account, and the circular motion of a satellite is obtained, see Figure 4. Figure 4. This is in line with the observations [3]. Beyond the particular case of the circular orbit, Newton was able to demonstrate that the general motion follows an ellipse or hyperbola beyond the exhaust velocity , and he was able to find the three laws previously established by Kepler based on precise measurements of the positions of the planets.

It happens that the force of gravity itself is proportional to the mass, so that the acceleration produced is independent of it: all bodies fall with the same acceleration in the same place. This equivalence between heavy mass and inertial mass was stated by Galileo, and illustrated by his famous experiments perhaps only imagined of falling objects from the Tower of Pisa. This was done again with great precision in the vacuum, avoiding air friction. As a demonstration for the public, a film comparing the fall of a hammer and a feather was made on the Moon during the Apollo XV mission [4] , and a similar experiment filmed on Earth in a huge vacuum chamber [5].

It is because of this principle of equivalence that all objects float in weightlessness in a satellite, each following exactly the same orbit around the Earth. These ultra-precise measurements are intended to test deviations from the principle of equivalence predicted by new gravitational theories. When an object falls, its kinetic energy increases, but its potential energy decreases as much, so that the total mechanical energy is preserved in the absence of friction. Mechanical energy conservation is becoming more common in the case of a satellite, but the expression of potential energy must then be modified to take into account the decrease in the force of gravity.

This potential energy depends only on the position of the object, so that it finds the same value after a revolution, and the kinetic energy also finds the same value, in accordance with the fact that the planetary movement persists indefinitely. Gravity potential is defined by dividing this potential energy by the mass of the object. This potential characterizes the gravitational field independently of the object orbiting it as long as the object is small and does not set the planet in motion by reaction.

The conservation of mechanical energy requires a particular form of the laws of force. Friction forces, unlike gravity, reduce mechanical energy, so that a satellite eventually falls into contact with the residual atmosphere. Heat essentially corresponds to the disordered kinetic energy of the gas molecules. Indeed, an object moving on an equipotential retains the same potential energy, and since its total energy is conserved, it cannot acquire or lose velocity under the effect of gravity alone.

Conversely, if the shape of the ocean deviates from an equipotential, surface water tends to flow to regions of lower potential until it fills them and reaches the equilibrium state where the surface is equipotential.