Doubt on this worked example from book by David Morin

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Homework Help Overview

The discussion revolves around a worked example from a classical mechanics textbook by David Morin, specifically focusing on the net moment about a pivot point in a problem involving torque and weight distribution along a stick. Participants are questioning the interpretation of the equations presented, particularly equation (2.57), and the reasoning behind the differentiation applied to it.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning, Assumption checking

Approaches and Questions Raised

  • The original poster attempts to understand the components of the net moment about the pivot, specifically questioning the exclusion of the moment due to the weight of the stick on the left side of the pivot. They also inquire about the reasoning for the second differentiation of equation (2.57). Other participants provide insights into the formulation of the torque contributions and the method of integration by parts used in the differentiation process.

Discussion Status

The discussion is active, with participants exploring different interpretations of the equation and the underlying principles of torque and weight distribution. Some guidance has been offered regarding the mathematical techniques involved, such as integration by parts, but there is no explicit consensus on the original poster's concerns.

Contextual Notes

Participants are navigating through the complexities of the problem, including the assumptions made about the weight distribution along the stick and the implications of the mathematical operations performed on the integral in equation (2.57). There is an acknowledgment of the need for clarity on the differentiation rule applied to the definite integral.

bksree
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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
 

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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.
 
Thank you.

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

TIA
 
cheers,

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

u= x-(xo+l) and dv = g*rho(x) dx
 

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