# D'Alembert's Principle:- A Falling Box

1. Jan 14, 2014

### Meekin

I've been asked to research D'Alembert's principle and solve a question. I've looked up quite a lot of different explanations on the internet of D'Alembert's Principle and I'm not quite grasping how to use it. I understand that you rearrange formula so that they equal 0 (e.g. f - ma = 0 or PE - KE = 0 ) so then you can treat them as static problems. I'm just really struggling on how to apply this to a question.

1. The problem statement, all variables and given/known data

A 25kg box is dropped from a height of 3.5m. Using D'Alembert's Principle work out the velocity reached immediately before hitting the ground.

(Gravity to be taken as 9.81ms-2)

2. Relevant equations

KE = (0.5 x (mass x velocity^2))

PE = mass x acceleration due to gravity x height

Final Velocity^2 = Initial Velocity^2 + (2 x Acceleration x Displacement)

3. The attempt at a solution

I already worked this out using the conservation of energy method and got an answer of Velocity = 8.287ms-1

I then tried playing around with it and realised i could treat it as though it was a box on a horizontal place with no friction

S = 3.5
A = 9.81ms-2
U = 0
V = ?

using v^2 = u^2 + 2as i got the same answer of 8.287ms-1.

This was a guess at a method because its the only other method I've managed to come up with but I'm fairly sure it doesn't use D'Alembert's Principle.

Any help would be appreciated. Thanks guys

2. Jan 14, 2014

### PhanthomJay

You are correct about D'Alemberts principle being, in its simplest form, a rearrangement of Newtons 2nd Law into an equation of dynamic equilibrium, that is , a rearrangement of F_net = ma into simply F_net - ma = 0, where the term " -ma " is a so-called pseudo force or inertial force which acts through the objects center of mass equal in magnitude to F_net and applied in the opposite direction. In your falling box example, the net force is mg acting down and the pseudo inertial force is ma acting up. So you write mg - ma = 0, from which a = g, and you solve for v using your kinematic equation to get of course the same result. The use of D'Alembert here is not very useful and can work against you if you are not careful.
The use of D'Alembert is more appropriate when dealing with torques , because you can then sum moments about any point = 0 for accelerating objects, instead of having to sum torques about the center of mass only if D'Alembert inertial forces are not used.