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Originally Posted by Juked07
1. Please characterize the effort that a helicopter expends to hover in place. Work is force dotted with distance, but distance here is 0, so it seems the usual physics quantification of work yields 0, which I find highly unsatisfactory. Perhaps impulse is relevant, but we don't typically use impulse to characterize effort and it has strangeness of it's own. Please explain.
Newton's Laws tell us that an object will accelerate if and only if the vector sum of forces on the object is non-zero.
In order to remain hovering (not accelerating), the helicopter must experience a vector sum of forces which is equal to 0 N. We know that gravity can be accurately modeled as a force in this circumstance. That force must be countered with an equal and opposite force to maintain a net force of 0 N.
In order to create this countering force, the helicopter does work to spin the rotors. The rotors have cross sections that are airfoils. When the air flows around the rotors, a pressure differential is created, with low pressure on top, and high pressure on the bottom. This creates lift, a force in opposition to gravity.
Consider a hovering rocket. It is stationary, so the work it experiences is 0 J. However, the gasses expelled from its jet experience a force as they are accelerated out the nozzle. The work done on the fuel creates the force which counters the force of gravity on the rocket.
Bear in mind that, in either case, a whole lot of energy is dissipated through friction, noise, heat, rotations in the air molecules, and linear momentum that is not directed in opposition to gravity.
Originally Posted by Juked07
2. Characterize the effort that a cyclist expends to climb a hill with constant gradient theta at two constant speeds, s1 and s2 where s2 > s1, assuming there is no energy lost to friction or air resistance. The force required at the two speeds is equal (only need to overcome another constant force, gravity, in both cases), so the total impulse over some time period t is equal. Yet at speed s2 the cyclist has clearly accomplished more than at speed s1. But work doesn't seem a fair measure either -- the impulse applied was the same!
Force per unit distance would be the same. So, as you pointed out, the energy (measured as impulse; J = Nm) would be the same.
However, the energy per unit time would be different.
Recall that power is energy/time (W = J/s). Moving at s2 would be consuming more power than moving at s1.
Revision: Oops, complete fail on my part. Impulse is NOT Fd (force x distance), it's Ft (force x time), and is NOT a measure of energy.
In your example, the force per distance is the same in both cases. The impulse delivered over a given time is, too. Since one rate traverses more distance than the other in the same time, it requires more energy in the same time. Since the acceleration is 0 m/s^2 in both cases, the net force on the cyclist must be 0 N, so the impulse is then 0 Ns.
Other than that, .
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