– Calculate the power needed to function the ski lift.
– Calculate the power needed to accelerate this ski lift in 5 s up to the speed of its operation.
The first objective of this question is to find the power required to operate the ski lift by first finding the work done as the power is equal to the work done per second. The power will be calculated with the equation as follows:
\[P=\frac{W}{t}\]
Where W is the work done while t is the time in seconds, the second objective is to find the power required to accelerate this ski lift.
This question is based on the application of Potential and Kinetic energy. Potential energy is energy that is stored and is dependent on the relative positions of multiple components of a system. In contrast, Kinetic energy is the object’s energy it has as a result of its motion.
Expert Answer
To calculate the power required to lift the ski lift, first, we have to calculate the work with the help of the formula:
\[W=mg \Delta z \]
The chairs are separated by $20m$ so at any moment the number of chairs lifted is:
\[N=\frac{1km}{20}=50\]
Next, we have to find the total mass with the formula:
\[m=N \times m_ {per chair}=50 \times 250=12500kg\]
\[W=12500 \times 9.81 \times 200 =24525000J\]
In order to calculate the power required to operate this ski lift, we first need to calculate the operation time.
\[t=\frac{d}{V}=\frac{1km}{10k}=360s\]
Power is defined as the work done per second, which is given as:
\[P=\frac{W}{t} = 68125W = 68.125kW\]
Then, we have to calculate the power needed to accelerate this ski lift in $5 s$ up to the speed of its operation.
The lift acceleration in 5 sec is:
\[a = \frac {\Delta V}{t}\]
where, $\Delta V$ is the velecoty change.
\[a=10 \times \frac {1000}{3600} – 0\]
\[=0.556 \frac{m}{s^2}\]
The amount of work required to accelerate the object is equivalent to the change in kinetic energy for an object or body and is calculated as:
\[W_a=\frac {1}{2}M(V_2^2 – V_2^1)kJ\]
\[=\frac{1}{2}(12500) \times (7.716)\]
\[=48225.308J\]
\[=48.225 kJ\]
Now the power required to accelerate the ski lift in 5s is given as:
\[W_a=\frac {W_a}{\Delta t}kW\]
\[=\frac{48.225}{5}\]
\[=9.645 kW\]
Now calculating the vertical distance traveled during the acceleration is given as:
\[h=\frac {1}{2}at^2 sin\propto \]
\[=\frac{1}{2} \times 0.556 \times 5^2 \times \frac{200}{1000}\]
\[=1.39m\]
Now the power due to gravity is given as:
\[W_g=Mg(z_2 – z_1)\]
\[=\frac {Mgh}{t} \]
\[=\frac {12500 \times 9.81 \times 1.39}{5}\]
\[=34.089 kW\]
Now the total power is given as:
\[W_{total}=W_a + W_g\]
\[=9.645 + 34.089\]
\[=43.734 kW \]
Numeric Result
The power required to operate the ski lift is $68.125kW$ while the power required to accelerate this ski lift is $43.734kW$.
Example
Find the power required to operate the ski lift which operates at a steady speed of $10km/h$ and a one-way length of $2km$ with a vertical rise of $300m$ and chairs are spaced $20m$ apart. Three people can be seated on each chair with the average mass of each loaded chair being $250kg$.
To calculate the power required to lift the ski lift, first, we have to calculate the work :
\[W=mg \Delta z \]
\[N=\frac{2km}{20}=100\]
Next, we have to find the total mass, which is given as:
\[m=N \times m_ {per chair}=100 \times 250=25000kg\]
\[W=25000 \times 9.81 \times 300 =73,575,000J\]
In order to calculate the power required to operate this ski lift, we first need to calculate the operation time.
\[t=\frac{d}{V}=\frac{2km}{10k}=0.2h=720s\]
Power is defined as the work done per second, which is given as:
\[P=\frac{W}{t} = 102187.5W \]