Negative vs Positive work on an escalator

[Matt Poirier]

Homework Statement

You are in a large store that has escalators connecting its floors. The stairs of each escalator move smoothly and steady either uphill or downhill as they carry passengers between floors.
You get off the "up" escalator on the second floor and board the "down" escalator. This escalator takes you from the second floor back to the first floor. Like the "up" escalator, the "down" escalator is 12 meters (39.4 feet) long. As before, the second floor is 4 meters (13.1 feet) above the first floor vertically. Let's continue to suppose that you weigh 600 Newtons (135 pounds-force). How much work does the "down" escalator do on you in carrying you from the second floor to the first floor?

I am confused as to whether or not the solution is positive or not. When the escalator moves you downward, doesn't it exert a force moving you downward making work positive? Or does the escalator always exert an upward force, and since you move downwards, work is negative?

Homework Equations

Work = force * distance

The Attempt at a Solution

-2400 joules, but I am not sure if it is negative.

 

[Chestermiller]

What is the direction of the force exerted on you, and what is the direction of your displacement?

 

[Matt Poirier]

Well I suppose the escalator exerts a support force upwards on you, but you move downwards. However, to move downwards, doesn't the escalator have to exert a force on you in the downward direction?

 

[Orodruin]

However, to move downwards, doesn't the escalator have to exert a force on you in the downward direction?

No, gravity is doing that for you. The elevator is stopping you from plunging into free fall.

 

[Matt Poirier]

No, gravity is doing that for you. The elevator is stopping you from plunging into free fall.

So it's negative?

 

[Orodruin]

So it's negative?

The work that the elevator does on you is negative. Hopefully it will exactly cancel the positive work that gravity does on you (or you will not stop).

 

[Matt Poirier]

The work that the elevator does on you is negative. Hopefully it will exactly cancel the positive work that gravity does on you (or you will not stop).

Does the same logic apply to escalators as they do elevators.

 

[Chestermiller]

Does the same logic apply to escalators as they do elevators.

Yes.

Here is another way of looking at it using unit vectors. The balance of the forces acting on you is given by:
$$N\mathbf{i_z}+mg(-\mathbf{i_z})=\mathbf{0}$$where N is the magnitude of the normal force exerted on you by the escalator, mg is the magnitude of the force exerted on you by gravity, and $\mathbf{i_z}$ is the unit vector in the upward direction. (I hope this makes sense to you so far)

Your downward component of your displacement is given by $\Delta z (-\mathbf{i_z})$, where $\Delta z$ is the magnitude of the downward displacement. If we take the dot product of this downward displacement vector with the force balance equation, we obtain:
$$-N\Delta z+mg\Delta z=0$$
The first term in the equation represents the work $W_N$ done by the escalator on you, and is negative. The second term in the equation represents the work $W_g$ done by gravity on you, and is positive. Note that: $$W_N+W_g=0$$Therefore,$$W_N=-W_g$$
Hope this helps.

 

[Matt Poirier]

Yes.

Here is another way of looking at it using unit vectors. The balance of the forces acting on you is given by:
$$N\mathbf{i_z}+mg(-\mathbf{i_z})=\mathbf{0}$$where N is the magnitude of the normal force exerted on you by the escalator, mg is the magnitude of the force exerted on you by gravity, and $\mathbf{i_z}$ is the unit vector in the upward direction. (I hope this makes sense to you so far)

Your downward component of your displacement is given by $\Delta z (-\mathbf{i_z})$, where $\Delta z$ is the magnitude of the downward displacement. If we take the dot product of this downward displacement vector with the force balance equation, we obtain:
$$-N\Delta z+mg\Delta z=0$$
The first term in the equation represents the work $W_N$ done by the escalator on you, and is negative. The second term in the equation represents the work $W_g$ done by gravity on you, and is positive. Note that: $$W_N+W_g=0$$Therefore,$$W_N=-W_g$$
Hope this helps.

Thank you so much!