A ski lift has a one-way length of 1 km and a vertical rise of200m. The chairs are spaced 20 m apart, and each chair can seatthree people. The lift is operating at a steady spedd of 10 km/h. Neglecting friction and ari drag ans assuming that the average massof each loaded chair is 250 kg, determine the power required tooperate the ski lift. Also estimate the power required toaccelerate this ski lift in 5 s to its operating speed when it isfirst turned

a) 68.125 KW

b) 43.04 KW

Explanation:

Distance = d = 1 km

Height = h = 200 m

Spacing between chairs = D = 20 m

No. of people per chair = 3

Speed = V = 10 km/h = 10000m/3600 s=2.8 m/s

mass of chair = 250 kg

Work to operate sky lift

W = mgh

Number of chairs any moment operational = N = 1 km/20=1000m/20=50

So, total mass of chairs = 50 * 250 = 12500kg

so, w = mgh=12500*9.8*200=2452500 j

Power is rate of work - we need time for operation time of lift

Time = t = distance/speedf = 1km / (10km/h) = 1km / (10 km/3600s) = 360s

So, Power P = Work/time=w/t=12500/360=68125 j/s=68.125 KW

Now calculating power for operating speed in 5 sec

We calculate accelleration=a for 5 sec

a = speed / time = V/52.8/5=0.28 m/sec2

for vertical acceleration we calculate θ angle first;

tanθ = height / distance = 200/1000 = 0.2

==> θ=11.34°

Vertical acceleration = a₁=a sinθ = 0.28 * sin 11.34=0.10835 m/sec2

to calculate height gained during startup use;

S=vit+1/2at2

here s=H

vi=0m/s

t=5 sec

==> H = 0+1/2a₁*t=0.5*0.10835 * 5²=0.5*0.10835*5*5=1.362 m

Total Work = mgH+0.5*m*V²=12500 (9.8*1.362+0.5*2.8*2.8) = 215240.56 j

Again power = work / time=215240.56/5=43048.112 W=43.04 KW