# Negative vs Positive work on an escalator

• Matt Poirier
In summary, when riding on an escalator, the work done on you by the escalator is negative, as the escalator exerts a support force upwards while you move downwards due to the force of gravity. This can also be seen through the use of unit vectors and the balance of forces acting on you. The work done by the escalator and the work done by gravity are equal in magnitude but opposite in direction, resulting in a total work of 0.
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.

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

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?

Matt Poirier said:
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.

Orodruin said:
No, gravity is doing that for you. The elevator is stopping you from plunging into free fall.
So it's negative?

Matt Poirier said:
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).

Orodruin said:
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.

Matt Poirier said:
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.

Chestermiller said:
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!

## What is the difference between negative and positive work on an escalator?

Negative work on an escalator occurs when the person is moving against the direction of the escalator, causing them to expend more energy. Positive work on an escalator occurs when the person is moving with the direction of the escalator, allowing them to conserve energy.

## Does negative work on an escalator burn more calories?

No, negative work on an escalator does not necessarily burn more calories. The amount of energy expended depends on factors such as the speed of the escalator, the person's weight, and the duration of the activity.

## Is it better to do negative or positive work on an escalator for exercise?

It is generally better to do positive work on an escalator for exercise, as it allows for a more efficient use of energy. However, a combination of both negative and positive work can provide a more well-rounded workout.

## Can negative work on an escalator cause injuries?

Yes, negative work on an escalator can potentially cause injuries such as muscle strains or sprains if the person is not careful or does not have proper form. It is important to use caution and maintain balance while using an escalator.

## What are some tips for incorporating negative work on an escalator into a workout routine?

If incorporating negative work on an escalator into a workout routine, it is important to start slow and gradually increase intensity. It is also recommended to maintain proper form, take breaks when needed, and listen to your body's limits.

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