Problem with energy conservation

[diegojco]

Suppose you have a teleski to climb a skiway of d=600m and θ=15º respect horizontal and μ_k=0.06 is kinetic friction coefficient in the cable. 80 skiers of 75kg (m=6000kg in total) each one wants to climb at the top at v=2.5m/s constant, What's the power that engine must provide in order to maintain this speed with that charge of skiers? Answer: 350KW

I have used: W_{external forces}=ΔE_mechanic+ΔE_thermic+ΔE_chemical=0, where ΔE_mechanic=(m*(vf)²/2)+m*g*(hf)-(m*(vo)²/2)-m*g*(ho)=m*g*(hf), since vf=vo y ho=0, and ΔE_thermic=f_k*d=μ_k*m*g*d*cos(θ), and ΔE_chemical is engine's energy to get the power, then:

P=-dE_chemical/dt

and -dE_chemical/dt=dE_mechanic/dt+dE_thermic/dt, hence:

P=m*g*v*(sin(θ)+μ_k*cos(θ))

P=46.61KW

This is not the answer, why?, please guide me and tell me what's the mistake.

 

[Sirus]

I'm getting something around 41kW as well. Keep in mind that the frictional force is directed against the motion of the gondola (teleski) up the cable, meaning the value of theta when multiplying by cos(theta) for friction is 0. The answer may be wrong; I don't know how they got an answer nine times larger than mine...

 

[e(ho0n3]

I'm getting 46 kW too. The answer of 350 kW must be wrong.

 

[Sirus]

diego, as far as method goes, I'm not sure what you did (probably ok since your answer looks right), but I usually approach this kind of problem using potential energy concepts. I set the time as one second, then find the vertical distance traveled given the speed provided, plug that into find the change in gravitational potential energy, and can therefore find power (you get J/s, which is Watts). Factor in non-conservative forces and angles when required. Just my method.