Question

A 80 kg skier grips a moving rope that is powered by an engine and is pulled at constant speed to the top of a 25 degrees hill. The skier is pulled a distance = 220 m along the incline and it takes 2.3min to reach the top of the hill.

If the coefficient of kinetic friction between the snow and skis is = 0.15, what horsepower engine is required if 30 such skiers (max) are on the rope at one time?

Answer 1

`P=28.085\,hp`

Step-by-step explanation:

Given that:

• mass of 1 skier,
  `m=80kg`

• inclination of hill,
  `\theta=25^(\circ)`

• length of inclined slope,
  `l=220m`

• time taken to reach the top of hill,
  `t=2.3 min= 138 s`

• coefficient of friction,
  `\mu=0.15`

Now, force normal to the inclined plane:

`F_N=m.g.cos\theta`

`F_N=80* 9.8* cos25^(\circ)`

`F_N=710.54\,N`

Frictional force:

`f=\mu.F_N`

`f=0.15* 710.54`

`f=106.58\,N`

The component of weight along the inclined plane:

`W_l=m.g.sin\theta`

`W_l=80* 9.8* sin25^(\circ)`

`W_l=331.33\,N`

Now the total force required along the inclination to move at the top of hill:

`F=f+W_l`

`F=106.58+331.33`

`F=437.91\,N`

Hence the work done:

`W=F.l`

`W=437.91* 220`

`W=96340.80\,J`

Now power:

`P=(W)/(t)`

`P=(96340.80)/(138)`

`P=698.12\,W`

So, power required for 30 such bodies:

`P=30* 698.12`

`P=20943.65\,W`

`P=(20943.65)/(745.7)`

`P=28.085\,hp`