An escalator in a shopping centre carries people up through a height h at a vertical speed of v, but you’re in a hurry when you step onto it, so you climb the moving steps (each with a height of hs) at a rate of n steps/s. Determine: the amount of chemical energy converted into mechanical energy by your leg muscles during your escalator ride, given that your mass is m; the work the escalator motor does on you; the total amount of work done by your legs and the escalator in getting you to the top
And: the total amount of work done by your legs and the escalator in getting you to the top.
Don't cheat. Do your own work
To determine the amount of chemical energy converted into mechanical energy by your leg muscles during your escalator ride, we need to calculate the work done by your legs.
1. First, calculate the distance you climb on the escalator:
- Since the escalator carries people up a height h at a vertical speed of v, the distance you climb is also h.
- Therefore, the distance climbed = h.
2. Next, calculate the work done by your legs:
- Work = Force × Distance
- The force exerted by your legs is equal to the weight of your body, which is given by the formula: weight = mass × gravity.
(Gravity is approximately 9.8 m/s²)
- So, Force = mass × gravity = m × 9.8
- Work = Force × Distance = m × 9.8 × h
The amount of chemical energy converted into mechanical energy by your leg muscles during your escalator ride is m × 9.8 × h Joules.
To determine the work the escalator motor does on you, we need to calculate the force exerted by the motor and multiply it by the distance traveled.
3. Calculate the force exerted by the motor:
- The force exerted by the motor equals the weight of your body plus the additional force needed to accelerate you vertically at the rate of v.
- The additional force is given by: additional force = mass × acceleration
(Acceleration = v² / (2 × hs) since the escalator moves at a speed of v and the height of each step is hs)
- So, additional force = m × (v² / (2 × hs))
- The total force exerted by the motor is: total force = m × 9.8 + m × (v² / (2 × hs))
4. Calculate the work done by the motor:
- Work = Force × Distance
- Given that the distance traveled is h, the work done by the motor is: Work = (m × 9.8 + m × (v² / (2 × hs))) × h
The total amount of work done by your legs and the escalator in getting you to the top is the sum of the work done by your legs and the work done by the motor:
Total Work = Work done by your legs + Work done by the motor
= m × 9.8 × h + (m × 9.8 + m × (v² / (2 × hs))) × h
Note: These calculations assume an idealized scenario without considering factors like efficiency, friction, or potential energy losses.