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

Answer:

$P=28.085\,hp$

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:

$F_N=m.g.cos\theta$

$F_N=80\times 9.8\times cos25^{\circ}$

$F_N=710.54\,N$

Frictional force:

$f=\mu.F_N$

$f=0.15\times 710.54$

$f=106.58\,N$

The component of weight along the inclined plane:

$W_l=m.g.sin\theta$

$W_l=80\times 9.8\times sin25^{\circ}$

$W_l=331.33\,N$

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:

$W=F.l$

$W=437.91\times 220$

$W=96340.80\,J$

Now power:

$P=\frac{W}{t}$

$P=\frac{96340.80}{138}$

$P=698.12\,W$

So, power required for 30 such bodies:

$P=30\times 698.12$

$P=20943.65\,W$

$P=\frac{20943.65}{745.7}$

$P=28.085\,hp$