Relationship between work and force

[sean39]

This is just a question that has been bothering me. I am not a student of physics, which is probably clear from my question. Hope it at least makes sense:

• You have a windmill with no friction.
  
• The wind is blowing at a constant speed causing the windmill to spin at 100 RPM.^(a)
  
• The windmill is hooked up to a pulley with no friction with a 1 kg weight attached.
  
• The energy transfer from the wind to the windmill to the pulley occurs with 100% efficiency.
  
• Now that the weight is attached, enough constant force is exerted by the wind on the windmill to raise the weight 1 m / sec. (1 joule of work)

What happens? Does the windmill slow down because of the weight? Seems like if the weight was attached while the windmill was already spinning, 100% of the force of the wind would be transferred to the pulley. But the windmill would just keep spinning anyway since there's no friction.
But if the windmill started out at rest when you hooked up the pulley, I don't know what would happen.
Would it take longer to lift the weight since the force of the wind is being used for both purposes, lifting the weight and accelerating the windmill blades? So is the weight raised at 0.5 m/ sec until the windmill gets up to 100 RPM, and then start going up at 1m / sec? Or something like that?

^(a) If the windmill has no friction, then it would spin perpetually at 100 RPM from a single gust. So I guess the amount of power exerted by that first gust to get the windmill from rest to a speed of 100RPM is all that matters. After that, the windmill would just spin at the same rate whether the wind continues to blow or not?

 

[andrewkirk]

Would it take longer to lift the weight since the force of the wind is being used for both purposes, lifting the weight and accelerating the windmill blades?

Yes, the windmill speed would initially be zero and then, once the blade lock was released to allow it to spin, its speed would increase. The acceleration would decline in order to make the speed asymptotically approach one that lifts the mass at 1m/s.

Initially all the work of the wind will go to accelerating the mass. As the speed increases, the proportion devoted to that task decreases as more of the work is devoted to working against gravity. The proportion devoted to acceleration will decline towards zero but never reach it. The mass's velocity will asymptotically approach 1m/s but never reach it.

 

[A.T.]

• You have a windmill with no friction.
  
• The wind is blowing at a constant speed causing the windmill to spin at 100 RPM.

...

So is the weight raised at 0.5 m/ sec until the windmill gets up to 100 RPM

With the weight, the windmill won't get to 100 RPM (nor will it asymptotically approach that speed). 100 RPM is the speed at which the wind exerts no torque on the windmill (equilibrium for a friction less windmill). With the weight it will asymptotically approach a speed where the aerodynamic torque balances the torque from the weight (less than 100 RPM).

 

[sean39]

Ok. Thank you.

What is the formula to determine the rate of acceleration? Does this mean with zero weight attached to the.blade it will asymptotically approach 100 rpm? What about with a 2 kg weight?

 

[A.T.]

What is the formula to determine the rate of acceleration?

https://en.wikipedia.org/wiki/Angular_acceleration#Equations_of_motion

Does this mean with zero weight attached to the.blade it will asymptotically approach 100 rpm?

Weight attached to the blade? I thought the weight is on a rope around a pulley.

What about with a 2 kg weight?

Will go slower than 1kg.

 

[Khashishi]

You are going to have some sort of gear ratio (probably adjustable) between the turbine and the weight. The gear ratio will lock a ratio between the angular velocity of the turbine and the velocity of the weight. If you have twice the weight with the same gear ratio, the turbine will be slower (or may even go backwards). But if you double the gear ratio, then you can have the turbine spin as fast as before (with 1kg weight), but the weight will move up half as fast.