Work and changing forces - Integration

Click For Summary

Discussion Overview

The discussion revolves around the concept of work done by a spring in mechanics, specifically focusing on the use of integral equations and the role of unit vectors in the calculation. Participants explore the mathematical representation of work, including the dot product and the interpretation of variables within the context of the problem.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Mathematical reasoning

Main Points Raised

  • One participant expresses confusion about the use of the unit vector i in the integral equations for work, questioning its role and the meaning of dr.
  • Another participant clarifies that i is a unit vector pointing in the +x direction and explains the dot product in this context, noting that both force and displacement are assumed to be along the x-axis.
  • A third participant elaborates on the integral for work, explaining that it involves analyzing vector force and the path of application, with dr representing small elements of displacement along the x-axis.
  • One participant emphasizes that since the problem involves a spring, it is reasonable to consider only one axis, while acknowledging that a more general case would require a three-dimensional approach.

Areas of Agreement / Disagreement

Participants generally agree on the interpretation of the unit vector and the application of the dot product in the context of work done by a spring. However, there is no consensus on the broader implications of the integral in three-dimensional scenarios, as some participants suggest that only one axis is necessary for this specific case.

Contextual Notes

There are assumptions regarding the directionality of forces and displacements, and the discussion does not resolve the potential complexities introduced by three-dimensional considerations.

urbanyoung
Messages
2
Reaction score
0
Hi,

I'm doing some mechanics study and one thing keeps confusing me. In the textbook (Serway/Jewett Physics for scientists and engineers, 7th ed) they introduce i into integral equations. I've put a picture of it below to save me from trying to type the symbols. The equation is for the work done by a spring.

j9arrl.png


I'm hoping someone can explain to me what is being done. I thought it would be dot product, but doesn't that introduce a cosine of the angle? I'm also not sure what the dr is, or well, what the r is.

Thanks.
 
Physics news on Phys.org
In this case, i is a unit vector (vector with magnitude 1) that points in the +x direction. i dot i = (1)(1)cos(0) = 1. In this case, the book apparently assumes that both the force and the displacement are along the x-axis.
 
The equation for work indeed involves the dot product. The integral used to calculate work does so by analyzing the vector force F and the path of its application. The path is split into tiny elements dr. For the particular application, namely compressing a spring, the force is acting along the x-axis, which is why dr changes to dxi. Similarly, the force compressing the spring is a function of the spring constant k and the amount the spring is compressed, xi. Remember, when a vector is dotted with itself, like i*i is in the integral, the resulting scalar quantity is the square of the length of the original vector, which is 1 for i. Since the angle between a vector and itself is zero, the cosine is equal to 1.
 
The equation is for the work done by a spring.

Since this is a spring its a pretty fair assumption that only one axis need be considered.
You would have to go into 3D for the general case.
 

Similar threads

High School Curious about Work and Energy Consumption
  • · Replies 8 ·
Replies
8
Views
2K
High School Work Done By Conservative Forces
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad Work-Energy Theorem for Moving Block and Spring System?
  • · Replies 4 ·
Replies
4
Views
3K
Undergrad What is the purpose of the vector α in the rotational work integral?
  • · Replies 3 ·
Replies
3
Views
1K
High School Work done when moving an object
  • · Replies 7 ·
Replies
7
Views
1K
Undergrad Work: Dot Product and Integral?
  • · Replies 7 ·
Replies
7
Views
4K
Undergrad Impulsive force and simple integration
  • · Replies 12 ·
Replies
12
Views
2K
Undergrad Why are there 2s -1 independent integrals of motion?
  • · Replies 4 ·
Replies
4
Views
2K
Graduate Principle of Virtual Work and the forces that DO NOT do work
  • · Replies 4 ·
Replies
4
Views
2K
High School Using energy principles to find equilibrium depth of floating object
  • · Replies 8 ·
Replies
8
Views
1K