Elevetor apparent weight, if normal force is less than the eight force what is the direction of velocity and acceleration?

Question

If you weigh yourself in an elevator, and the normal force is less than the weight..

a) What is the direction of velocity
b)What is the dorection of acceleration

Answer 1

Hi Essie!

 

Let's take a look at this through Newton's 2nd law of motion:

 

F_met = ma

 

In this case, for the person in the elevator, there are two principal forces at work:  gravity, downward (which I will call the negative direction); and the normal force of the elevator floor on the person, upward (positive).  So we can write Newton's 2nd law as:

 

F_net = N - mg = ma

 

The problem describes a situation where the normal force is less than the weight (N < mg).  As a result, N - mg is negative, meaning the acceleration has to be negative, by Newton's second law.  So, the acceleration is downward (since I defined down to be negative).

 

However, one bedrock truth about acceleration is that the direction of acceleration is unrelated to the direction of velocity without more information.

 

For example, imagine this elevator is two different situations:

 

1.  Speeding up in the downward direction

 

In this case, we start with some negative velocity v_o, and, at a later time, have a negative velocity v with a greater magnitude, such that v-v_o < 0 (if it is easier to imagine with numbers, take v_o = -2 m/s and v = -5 m/s; then v-v_o = -5-(-2) = -3 m/s).  With this case, we have:

 

a = (v-v_o)/Δt,  and, since v-v_o < 0, the acceleration is negative, and the velocity is negative the whole way

 

2.  Slowing down in the upward direction

 

In this case, we start with some positive velocity v_o, and end up with a final positive velocity v that is smaller in magnitude (e.g., v_o = 5 m/s, and v = 2 m/s).  In this case, v-v_o is also negative (e.g. 2 - 5 = -3 m/s), and:

 

a = (v-v_o)/Δt; since v-v_o < 0, the acceleration is negative, though the velocity is positive the whole way

 

This is an example of how a negative acceleration can exist with the elevator moving in EITHER direction.  Unlike velocity, whose sign does tell you about the direction of motion, the acceleration does not relate to the direction of motion without more information.  Some examples of information that, if we were told it, would help determine the direction of velocity, are:

 

* the initial velocity of the elevator (including a statement that it starts from rest, which would mean it would have to be moving downward and thus have negative velocity)

 

* a statement of whether the elevator is speeding up or slowing down, because the case of speeding up (it turns out) means acceleration and velocity point in the same direction, and the case of slowing down means they point opposite directions.

 

So, without more information, all we can say is:  the elevator either has positive velocity and is slowing down, or negative velocity and is speeding up.  Both would produce the reduced apparent weight that is given.

 

Remember, net force is not NECESSARILY related to the direction of motion.  Net force is related (by Newton's 2nd law) to the direction of acceleration.