Power related to the EMF of an adjustable coil in a magnetic field

[garthenar]

Homework Statement

You have designed a new type of exercise machine with an extremely simple mechanism (Figure 1). A vertical bar of silver (chosen for its low resistivity and because it makes the machine look cool) with length 3.0 m is free to move left or right without friction on silver rails. The entire apparatus is placed in a horizontal uniform magnetic field of strength 0.25 T. When you push the bar to the left or right, the bar's motion sets up a current in the circuit that includes the bar. The resistance of the bar and the rails can be neglected. The magnetic field exerts a force on the current-carrying bar, and this force opposes the bar's motion. The health benefit is from the exercise that you do in working against this force.

A) Your design goal is that the person doing the exercise is to do work at the rate of 25 watts when moving the bar at a steady 2.0 m/s. What should be the resistance R?
Express your answer using two significant figures.

B) You decide you want to be able to vary the power required from the person, to adapt the machine to the person's strength and fitness. If the power is to be increased to 50 W by altering R while leaving the other design parameters constant, should R be increased or decreased?

C) You decide you want to be able to vary the power required from the person, to adapt the machine to the person's strength and fitness. If the power is to be increased to 50 W by altering R while leaving the other design parameters constant. Calculate the value of R for 50 W

D) When you start to construct a prototype machine, you find it is difficult to produce a 0.25-T magnetic field over such a large area. If you decrease the length of the bar to 0.20 m while leaving B, v, and R the same as in part A, what will be the power required of the person?

Relevant Equations

P=VI

V=Emf =(d(ΦB))/dt

W=F*d

I'm already stuck on A. I'm hoping once I figure that out the rest will just fall into place but be prepared for this to take awhile.

I understand how to use Faraday's law to get the current or voltage of the system based off the movement of the bar but I have no Idea how to relate the rate of work, or work into the mix.

I'm used to work being force times distance. Then would power be... I can't figure out how to relate velocity into the mix either...

if V=ds/dt and we have a "rate of 25W" which I'm assuming is 25[W]/s and P=dWrk/dt = Fs/t = ... I have no Idea

My work is just nonsensens like that over and over, and is a total mess so I have not included it.
What I need right now is for someone to help me figure out how to approach this scenario. What am I missing?
And no, I could not find a similar problem or example in the textbook.

 

[mjc123]

What has your question to do with the title of your post?

 

[DEvens]

So you have an equation that gives you voltage as the time derivative of flux. So how much flux is getting moved from one side of the bar to the other each second? Remember what the T means in the definition of flux.

https://en.wikipedia.org/wiki/Tesla_(unit)
Then you have a voltage. And you have a desired power for that to produce. What resistance does that voltage have to be going through to get that power?

 

[garthenar]

What has your question to do with the title of your post?

Sleep deprivation. I'll see if I can fix it.

 

[garthenar]

So you have an equation that gives you voltage as the time derivative of flux. So how much flux is getting moved from one side of the bar to the other each second? Remember what the T means in the definition of flux.

https://en.wikipedia.org/wiki/Tesla_(unit)
Then you have a voltage. And you have a desired power for that to produce. What resistance does that voltage have to be going through to get that power?

Thank you, Once I realized that I would taking the derivative of (2[m/s])t in the Emf equation everything fell together.