Question

A major department store has just installed three new escalators. Each one is designed to carry 120 people, each of average mass 70kg. per min to a floor 10.0m higher. The engine which drives these escalators is 65% energy efficient. What?

What power must the engine develop?????

Answer 1

`"P"("input")=2.1 xx 10^(4) color(white)(l) "W"`

Explanation:

From the definition of average power

`"Power" = "Work" / "Time"`

Meaning that the energy input to the escalator in one minute (`60 color(white)(l) "seconds"`) divided by time shall give its average power input.

The effective work the escalator does in a one-minute interval is equal to the passengers' gain in gravitational potential energy in that amount of time. Taking the gravitational field strength `g=9.8 color(white)(l) "N" * "kg"^(-1)`

`W("output") = Delta "PE" = m*g* Delta h`
`color(white)(W_"output" = Delta "PE" ) = (120*70 color(white)(l))color(red)(cancel(color(black)("kg")))*9.8 color(white)(l)"N"*color(red)(cancel(color(black)("kg"^(-1)))) cdot 10.0 color(white)(l)"m"`
`color(white)(W_"output" = Delta "PE" ) = 8.2 xx 10^(5) color(white)(l)"N"*"m"`
`color(white)(W_"output" = Delta "PE" ) = 8.2 xx 10^(5) color(white)(l)"J"`

The escalator has an efficiency of `65 %`, meaning that it converts only `65%` of the energy input into the desired form of work. That is:

`("W"("output"))/("W"("input"))=65%`
`"W"("input")=("W"("output") )/(0.65)=1.3 xx 10^(6) color(white)(l) "J"`

Note that `1 color(white)(l)"Watt"=(1 color(white)(l) "Joule")/(1 color(white)(l)"second")`

`"P"("input")=("W"("input"))/(t)=(1.3 xx 10^(6) color(white)(l) "J")/(60 color(white)(l) "s")=2.1 xx 10^(4) color(white)(l)"W"`