Doubt on this worked example from book by David Morin

In summary, the conversation discusses a worked problem in the Classical Mech book by Morin and the confusion about equation 2.57 being the net moment about the pivot. The speaker explains that the first rectangle of the integral represents the weight of a small piece of the stick at a distance from the pivot, and the differentiation is used to find rho(x).
  • #1
bksree
77
2
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
 

Attachments

  • Doc1.doc
    146.5 KB · Views: 212
Physics news on Phys.org
  • #2
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 hight 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.
 
  • #3
Thank you.

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

TIA
 
  • #4
cheers,

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

u= x-(xo+l) and dv = g*rho(x) dx
 
  • #5

I understand your confusion and appreciate your attention to detail in this worked example. Let me provide some clarification on the equations and their meaning.

Firstly, equation (2.57) represents the net moment about the pivot, as it takes into account the weight of the entire stick, both to the right and left of the pivot. This is because the weight of the stick on the left side also contributes to the overall moment, even though it is not directly acting on the pivot.

The equation is differentiated a second time to solve for the angular acceleration of the stick, which is necessary to determine its rotational motion. This is a common practice in mechanics problems, as differentiation often allows us to solve for unknown quantities and understand the dynamics of a system.

I hope this explanation helps to clarify any doubts you may have had about the worked example. If you have any further questions or concerns, please do not hesitate to ask. As scientists, it is important to question and seek understanding in order to further our knowledge and advance in our fields. Thank you for your curiosity and engagement in this topic.
 

1. How can I be sure that the worked example in the book is correct?

The best way to confirm the accuracy of the worked example is to double-check the calculations and reasoning used to arrive at the solution. You can also compare the example to similar problems or consult other resources for additional explanations or solutions.

2. What if I don't understand a step in the worked example?

If you are having trouble understanding a particular step, try breaking it down into smaller parts and working through each one individually. You can also reach out to a classmate, teacher, or online forum for clarification or additional explanations.

3. Are there any common mistakes that may occur in this worked example?

There may be some common mistakes that could occur, such as typos or errors in calculations. It is always a good idea to review your work and check for any potential mistakes before moving on to the next step.

4. Can I use the worked example as a template for solving similar problems?

Yes, the worked example can serve as a guide or template for solving similar problems. However, be sure to understand the underlying concepts and reasoning behind each step rather than simply copying the solution.

5. What if I get a different answer than the one shown in the worked example?

If you get a different answer, it could be due to a mistake in your calculations or a different approach to solving the problem. Double-check your work and if you still get a different answer, try approaching the problem from a different angle or consulting other resources for additional help.

Similar threads

  • Introductory Physics Homework Help
Replies
7
Views
1K
  • Introductory Physics Homework Help
Replies
2
Views
2K
  • Mechanics
Replies
2
Views
1K
  • Advanced Physics Homework Help
Replies
2
Views
4K
  • Introductory Physics Homework Help
Replies
11
Views
1K
  • Introductory Physics Homework Help
Replies
2
Views
3K
  • Introductory Physics Homework Help
Replies
17
Views
3K
  • Introductory Physics Homework Help
Replies
1
Views
1K
  • Introductory Physics Homework Help
Replies
1
Views
1K
Replies
6
Views
4K
Back
Top