What is d'Alembert's Principle?

This post is really more to do with statics, surprisingly, and concerns a way of tackling dynamics problems that was devised by a French scientist called d’Alembert. At the time of its introduction it was very difficult to understand but nowadays we are all familiar with some of its applications in space travel.

Suppose we consider the case of astronauts being launched into space on a space shuttle. During the launch when the shuttle is vertical and accelerating rapidly up through the atmosphere with the rockets at full blast, the astronauts feel as though they have suddenly become extremely heavy; they cannot raise their arms from the arms of the seat, they cannot raise their heads from the headrests and the skin on their faces is pulled back tight because it is apparently so heavy.

We know that all this is because the shuttle is accelerating so fast that the astronauts are subjected to what are known as ‘g-forces’, the sort of forces you feel on a roller coaster as it goes through the bottom of a tight curve.

However, if you were in the space shuttle and the windows were blacked out so that you could not see any movement, you could be excused for thinking that the astronauts really were being subjected to some large extra force acting in the opposite direction to the acceleration.

For an astronaut of mass m experiencing an acceleration of a directly upwards, the extra imaginary force would be F = ma directly down, according to Newton’s second law. From that point on, any engineering dynamics problem concerning the astronaut, such as working out the reaction loads on the seat anchor points, could be carried out as a statics calculation with this imaginary force included.

This is d’Alembert’s principle.

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