sarthak sharma said:
i just wanted to know that how work done is related to distance because as far as my knowledge goes we relate work done with displacement...
if there is anything different for this case can pls explain it a bit more...
Distance is referred to the magnitude of displacement. But in this particular case it's necessary to talk about distance traveled. It was argued that the force exerted by the man is nonconservative, and hence the path (not just the ##\Delta d## between two points) must be taken into account. Consider that:
- In the best scenario, the work done by this force, ##W_{nc}##, will be equal to the ##PE_g##. Otherwise it will be greater.
- The distance traveled (let call it ##d##) is ##d =\bar{v}t##, where ##\bar{v}## is the average velocity. Then ##t = \frac{d}{\bar{v}}##
The net, average velocity was ##v-u##. It follows that:
##W_{nc} \geq PE_g##
##F_{nc}d \geq mgh##
##F_{nc}\frac{d}{v-u} \geq mg\frac{h}{v-u}## <-- We want to find for how long the force was exerted (without writing ##t## explicitly). Look at the right-hand side. You end up with units of momentum. ##\frac{d}{v-u}## is the time he was running in the escalator (another way of viewing this is by looking at the step the man took, as collinsmark says).
##F_{nc}\frac{d}{v-u}v \geq mg\frac{h}{v-u}v## <-- Now we want to find the distance traveled by the man. We multiply the time he spent running (written in terms of ##d## and ##v-u##) by the (average) velocity of the man ##v##.
The equation is just telling us that there is no way that the work done is equal to ##mgh##. You need to either climb straight up a height ##\frac{h}{v-u}v## at constant velocity, or to consider that the work done also went to other forms of energy.
sarthak sharma said:
pls can u explain that which non conservative force is in action here...?
The force the man exerts going up the escalator. Let suppose it's a conservative force. Then, Conservation of Mechanical Energy should apply:
##KE_i + PE_i = KE_f + PE_f##
##KE_i = KE_f + PE_f ## <- take ##h_i = 0##
##KE_i = PE_f## <-- Suppose at the top his ##v = 0##.
Now pick an arbitrary mass (of a person), a fixed ##h## and vary ##v_0##. The equation doesn't hold. By Conservation of Energy that can't be right. The difference energy input/output must go somewhere. Then we conclude that the force exerted should be nonconservative:
##W_{nc}= PE_f + OE_f##
##F_{nc}d = PE_f + OE_f## <- some of the energy going up goes into other form of energy (OE).
If ##F_{nc}d = PE_f##, means that ##OE_f=0##, which in turn means that the man doesn't feel tired, hot, doesn't need to take a breath, etc (is a superhero).
