an escalator carries you from one level to the next in the airport terminal. the upper level is 4.5 m above the lower level, and the length of the escalator is 7 m. your mass is 60 kg.
how much work does the up escalator do on you when you ride it from the lower level to the upper level?
how much work does the does the down escalator do on you when you ride it from the upper level to the lower level?
Work = potential energy of you
Work =mass *9.8 m/s *Height
Work = 60kg *9.8*4.5
Work =2646 J =2.646 kJ
work=force dot distance where dot means the dot product. Dot product means the component in the same direction. Here force is mg, so the distance in that direction is 4m
Notice in the second, the movement is downward, gravity is doing the work, the escalator is absorbing it, hence the negative sign.
Well, let's calculate the work done by both the up escalator and the down escalator, shall we?
First, let's tackle the up escalator. The work done by the up escalator is equal to the change in potential energy. We can calculate the potential energy using the equation:
Potential Energy = mass * acceleration due to gravity * height
In this case, the height is 4.5 m. The acceleration due to gravity is approximately 9.8 m/s^2. And your mass is 60 kg.
So, the potential energy is:
Potential Energy = 60 kg * 9.8 m/s^2 * 4.5 m
Now, since the work done is equal to the change in potential energy, we can conclude that the up escalator does work equal to the potential energy, which is:
Work done by up escalator = 60 kg * 9.8 m/s^2 * 4.5 m
And to answer the second question, when you ride the escalator down, the escalator is doing negative work on you, which means it is actually reducing your potential energy. So, the work done by the down escalator is the negative of the potential energy:
Work done by down escalator = -60 kg * 9.8 m/s^2 * 4.5 m
So, there you have it! The work done by the up escalator is 60 kg * 9.8 m/s^2 * 4.5 m, and the work done by the down escalator is -60 kg * 9.8 m/s^2 * 4.5 m.
To calculate the work done by the escalator on you, we need to use the formula:
Work = Force × Distance × cosθ
where Force is the vertical force applied by the escalator, Distance is the distance traveled along the inclined plane, and θ is the angle between the force applied by the escalator and the direction of motion.
For the upward escalator:
1. Calculate the force applied by the escalator:
The force applied by the escalator is equal to the component of your weight that acts parallel to the inclined plane. Since you are in equilibrium while moving, this force equals your weight:
Force = m × g
where m is your mass and g is the acceleration due to gravity (9.8 m/s²).
2. Calculate the distance traveled along the inclined plane:
The distance traveled along the inclined plane is the same as the length of the escalator, which is 7 m.
3. Calculate the angle θ:
Since the escalator is moving upward, the angle θ is the angle between the inclined plane and the vertical direction, which is 90 degrees.
Now we can calculate the work done by the upward escalator:
Work = Force × Distance × cosθ
= m × g × 7 × cos90°
= m × g × 7 × 0
= 0
Therefore, the upward escalator does zero work on you.
For the downward escalator, the process is similar:
1. The force applied by the escalator is still your weight, so the force is the same as before: m × g.
2. The distance traveled along the inclined plane is still the length of the escalator, which is 7 m.
3. The angle θ is now 180 degrees since the escalator is moving downward.
Now we can calculate the work done by the downward escalator:
Work = Force × Distance × cosθ
= m × g × 7 × cos180°
= m × g × 7 × (-1)
= -m × g × 7
Since the work is negative, it means that the downward escalator does negative work on you, which is equivalent to you doing work on the escalator.
Therefore, the downward escalator does -m × g × 7 J of work on you.