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D’alembert’s principle states that the sum of the differences between the forces acting on a mass particle and the rate of change of momentum of the system itself along any virtual displacement is zero.
∑i(Fi−miai)δri=0 Fi=Net force acting on ith particle
mi=Mass of ith particle
ai=Acceleration of ith particle
ri= Virtual displacement of ith particle
Note:-In many textbooks, it is given as ∑F−ma=0, that is a special case (refer Wikipedia).
Note: Please ask your profs regarding what to write in the exams.
Example 1:-
Consider 2 teams playing tug-of-war. A box of mass ‘m’ is attached to a rope at two opposite places. Team A pulls the box with force FA and Team B pulls it with force FB. The force of Team A is more, so the box is accelerated towards Team A.
By D’alembert’s principle,
[FA+FB]−ma=0
Example 2:-
A box of mass ‘m’ is kept on an inclined surface with co-efficient of friction ‘µ′ and is accelerated downwards.
By D’alembert’s principle (along x-axis)
[-f+mg \sin\theta ]-ma=0 \\ i.e. \space -μ(mg \cos\theta )+ mg \sin\theta -ma=0