I have a question with process when using D'Alembert/Virtual Work.(adsbygoogle = window.adsbygoogle || []).push({});

I have the example question:

A uniform plank of mass M is leaning against a smooth wall and makes an angle α with the smooth floor. The lower end of the plank is connected to the base of the wall with a massless inextensible string. A monkey of mass m starts climbing on the plank. If the maximum tension force on the string before it breaks is T, up to what maximum height can the monkey climb to?

M =mass of plank

m = mass of monkey

g = force due to gravity

2b = length of plank

a = distance monkey is up plank

x = length of string, distance between wall and where plank meets floor

y = height of center of mass of plank

z = height of center of mass of monkey

Assuming a static case just before the string breaks we have:

x = 2bcosΦ

y = bsinΦ

z = asinΦ

and

δW = Tδx + Mgδy + mgδz

If I set δW to 0 and take δx, y and z to be the derivatives of x, y and z with respect to Φ I get

0 = MgbcosΦdΦ + mgacosΦdΦ - T2bsinΦdΦ

a = (2bTtanΦ-Mgb)/mg

Which is a similar process to what I would use in the more simple example without the monkey however I saw this http://imgur.com/FqjZv which seems to take the partial derivatives of each of the independent variables then sum them to get the virtual displacements. That second example then goes on to set each displacement to zero and get equations of motion. In comparison in the first example I simply took the derivative with respect to Φ which solves simply.

In the monkey example z has a in it which I think could be seen as an independent variable so:

Have I done the monkey question right?

In the linked example is the person taking the partials of each θ then adding the results as I suspect?

How do I know when/where to use these seemingly different techniques?

Thanks.

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

2. Relevant equations

3. The attempt at a solution

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# D'Alembert/Virtual Work Problem

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