Why is it easier to undo a bolt with a long lever

[climbon]

Question about moments and torques.

I understand that torque is defined as τ=Fs and understand where this comes from mathematically but am wondering why does nature make it easier to put a greater torque on an bolt with a longer lever.

The only real explanation I can think of is by increasing the length of the arm, you are increasing the radius of the turning circle (of your lever arm). By doing this you are increasing the distance moved of your hand (circumference of circle) and so increasing the work done (W=Fs) and so more "energy" goes into turning the bolt with a longer arm?

Thanks!

 

[Nugatory]

The only real explanation I can think of is by increasing the length of the arm, you are increasing the radius of the turning circle (of your lever arm). By doing this you are increasing the distance moved of your hand (circumference of circle) and so increasing the work done (W=Fs) and so more "energy" goes into turning the bolt with a longer arm?

That's pretty much it, but you could also think about from the bolt's point of view. For the bolt, the lever arm is one-half its diameter. When you increase your lever arm, the forces holding the bolt frozen in its socket have relatively less leverage to resist the force you're applying.

 

[CWatters]

I agree. Think about levers in general. It's not just the length of the lever that matters, the ratio of the two parts of the lever are important.

By doing this you are increasing the distance moved of your hand (circumference of circle) and so increasing the work done (W=Fs) and so more "energy" goes into turning the bolt with a longer arm?

I think it's more constructive to think of the forces & torques involved rather than the work/energy. It's probably best to think of the work/energy required to undo the nut as a constant. It's the same nut and bolt regardless of how long the wrench is!

Basically the lever provided by a longer wrench means your arm has to apply a lower force on the end of it... but in the end the work done (work = force * distance) is the same. The longer wrench just trades force for distance.