Doubt on this worked example from book by David Morin

[bksree]

Attached is a worked problem in the Classical Mech book by Morin.

I don't understand how eqn (2.57) is the net moment about the pivot. Isn't it only the moment of the weight of the portion of the stick to the right of the pivot ? What about the moment due to the weight of length 'l' on left of pivot ?

Why is the equation 2.57 differntiated the second time ?

Please explain.

TIA

 

[eczeno]

the formula appears correct to me. maybe this will help. think of the integral as a Riemann sum, a bunch of rectangles under the function rho(x)*(x-(xo+l))*g . now, think of the first rectangle (and let there be lots of rectangles). this will have a height rho(xo)*l*g and width dx, so the torque contribution from the first dx of the stick will be: rho(xo)*l*g*dx. this is correct! the weight of that little piece is rho(xo)*g*dx and it is a distance l from the pivot.

as for why he differentiated again, i don't see any other way to find rho(x). he simply turns a hard integral problem into an easy differential equation.

 

[bksree]

Thank you.

BTW, what is the rule being used to differentiate the definite inegral in eqn 2.57 ?

TIA

 

[eczeno]

cheers,

he does integration by parts on 2.57 before taking the derivative:

u= x-(xo+l) and dv = g*rho(x) dx