Problems with work and F*dr

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You need to use the dot product to correct the expressionIn summary, the conversation discusses a semi-circle defined by a given function and an attractive force that acts on the semi-circle. The work done against the force when a particle moves along the semi-circle from θ = 0 to θ = π is calculated using the equation W = -F0Rπ. However, an error may have been made in interpreting the meaning of dr in this calculation.
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A semi-circle is given by the function [tex]y=R cos(\frac{x\pi}{2R})[/tex]. An attractive force F at [tex]x = R[/tex] and [tex]y = 0[/tex] such that [tex]F(x, y) = F(R, 0) = 0 [/tex] and [tex]F(x, y) = F(-R, 0) = F_{0} [/tex]. This force along the semi-circle, relative to its source, is given by [tex]F(\theta) = F_{0} sin(\theta/2)[/tex]. What amount of work is done against the force when a particle moves along the circle from [tex]\theta = 0[/tex] to [tex]\theta = \pi[/tex].
To find the work I have the equation
[tex]W = - \int^{\pi}_{0} F dr[/tex]
For the force I have the equation
[tex]F= F_{0} sin(\theta/2)[/tex]
For the distance from the particle to the orgin of the force I have
[tex]2 R sin(\theta/2)[/tex]
so for dr I have
[tex]2 R sin(\theta/2) d\theta[/tex]
Putting this all tegether I get
[tex]W = - \int^{\pi}_{0} F_{0} sin(\theta/2) 2 R sin(\theta/2) d\theta[/tex]
This rearanges to
[tex]W = -F_{0} R \int^{\pi}_{0} \left[1-cos(\theta) \right] d\theta[/tex]
Integrating I get
[tex]W = -F_{0} R \left[ \theta-sin(\theta) \right]^{\pi}_{0}[/tex]
This works out to
[tex]W = -F_{0} R \pi[/tex]
The answer I am suppose to get is
[tex]W = -F_{0} R[/tex]
What did I do wrong?
 
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  • #2
The question doesn't make very much sense. Could you please reproduce the question exactly as it appears in the text, and also provide the name of the text ?
 
  • #3
I rewrote the problem a little bit. I'm afraid it's still a little unclear without the diagram though. This question is not from a textbook.
 
  • #4
I think the mistake you are making is in interpreting what dr refers to. This is an elemental distance traveled by the particle (along the semicircular path) - it is not an element of the distance of the particle from the origin. Also keep in mind that the angle between F and dr changes with the motion of the particle.
 

1. What is "Problems with work and F*dr"?

"Problems with work and F*dr" refers to the challenges and issues that arise in the workplace, specifically related to the Franklin D. Roosevelt administration and the New Deal policies that were implemented during his presidency.

2. What were some common problems with work during the New Deal era?

During the New Deal era, some common problems with work included low wages, long working hours, unsafe working conditions, and discrimination against certain groups of workers, such as women and people of color.

3. How did the New Deal address these problems with work?

The New Deal implemented various policies and programs to address these problems with work. For example, the Fair Labor Standards Act established a federal minimum wage and maximum work week, while the National Recovery Administration promoted fair labor practices and collective bargaining rights for workers.

4. Were there any criticisms of the New Deal's approach to addressing problems with work?

Yes, there were criticisms that the New Deal did not do enough to address the problems with work. Some argued that the policies did not go far enough to protect workers' rights and improve working conditions, while others believed that the New Deal favored big businesses over workers.

5. How do the problems with work and F*dr relate to current workplace issues?

Many of the problems with work that existed during the New Deal era still persist in modern workplaces, such as low wages, long hours, and discrimination. However, there have also been advancements in labor laws and workers' rights since then. It is important to continue addressing these issues in order to create fair and safe working environments for all individuals.

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