F = ma for a lift (elevator) carrying passengers

[hendrix7]

Summary:: What constitutes the 'body' when applying equation of motion?

I was solving this problem:
'A woman of mass 60 kg is in a lift of mass 250 kg which is accelerating downwards at 3.2 m^-2. Find the tension in the cable of the lift.'
when I realized that I'm not sure what constitutes the 'body' when applying F = ma and, in particular, what should be included in mass (m)? For example, I can't resolve the forces on the lift (F in the equation) without including the mass of the woman (m) in the equation. As far as I can see, I have to treat both the lift and the woman as one body but what is the general rule for this? How will I know when I have to include more than one object in the 'body' when using the equation of motion?

 

[kuruman]

Since you are looking for the tension, the body or system must be something that is connected to the cable. Since you want to use Newton’s second law, the system must be something whose mass is given. Since the woman and the lift accelerate as one, you must include them both otherwise you would have to worry about the normal force exerted by the woman’s feet on the lift. If you were also asked to find the normal force, then you would have to draw two diagrams. Do you see how one sorts it out? Your choice was correct.

 

[jbriggs444]

As far as I can see, I have to treat both the lift and the woman as one body but what is the general rule for this?

You could solve this by treating the lift and the woman separately. However, it is more difficult than the approach that you have already selected. Let me walk you through...

One starts with a free body diagram for each free body in the problem: The elevator and the woman. This is a mental aid to help you identify the external forces that act on each.

Select a coordinate axis (up for positive, down for negative).

Consider the woman. Subject to a downward force, $F_{gw}$ from gravity on the woman, $F_{gw} = -mg$. Also subject to an upward force from the lift floor, $F_{lw}$. The value of this second force is unknown. However, we also know that both woman and elevator are accelerating downward at a rate $a$. This acceleration is given as -3.2 m/sec². [You quoted the wrong units in the problem statement]

From this information, you can apply Newton's second law for the woman and compute $F_{lw}$

Now that you know $F_{lw}$, by Newton's third law you also know $F_{wl}$.

Now repeat the analysis for the elevator alone. You have three external forces this time. One from gravity, one from the woman's feet on the lift floor and one from tension. You want to determine the one from tension. But you know the other two and the resulting acceleration. So you apply Newton's second law and solve for $F_T$. At no point does this approach require you to to add up the masses of the lift and of the woman.

[You do end up adding the gravitational forces on each, so your mathematical legerdemain has not saved any work].

 

[PeroK]

Summary:: What constitutes the 'body' when applying equation of motion?

As far as I can see, I have to treat both the lift and the woman as one body but what is the general rule for this? How will I know when I have to include more than one object in the 'body' when using the equation of motion?

It's a good question. Consider also that the cable may be connected to a part of the lift (a bracket, perhaps) with a mass of $25kg$. Do we consider only that $25kg$ and not all of the lift?

The answer is that we can consider everything that is moving together as the mass of the system that the cable supports. The justification is that internal forces must cancel (Newton's third law). We could make it a multi-body problem, but as internal forces cancel, we must get the same answer as considering the bracket, the main body of the lift and the woman as a single massive object.

If the woman is jumping up and down in the lift, or otherwise moving relative to the lift, then we would have to consider the forces on the lift and woman separately in the free-body diagram.

 

[A.T.]

As far as I can see, I have to treat both the lift and the woman as one body

You don't have to, but you can in this case and it simplifies the computation.

but what is the general rule for this?

There is none. You have to decide how you define the bodies based on the information given and question asked. Just like with choosing the reference frame for analysis, you might have multiple possible choices. But some might result in simpler math.

How will I know when I have to include more than one object in the 'body' when using the equation of motion?

Imagine for example the question also asks about the force between the lift floor and the woman's feet.

 

[hendrix7]

You could solve this by treating the lift and the woman separately. However, it is more difficult than the approach that you have already selected. Let me walk you through...

One starts with a free body diagram for each free body in the problem: The elevator and the woman. This is a mental aid to help you identify the external forces that act on each.

Select a coordinate axis (up for positive, down for negative).

Consider the woman. Subject to a downward force, $F_{gw}$ from gravity on the woman, $F_{gw} = -mg$. Also subject to an upward force from the lift floor, $F_{lw}$. The value of this second force is unknown. However, we also know that both woman and elevator are accelerating downward at a rate $a$. This acceleration is given as -3.2 m/sec². [You quoted the wrong units in the problem statement]

From this information, you can apply Newton's second law for the woman and compute $F_{lw}$

Now that you know $F_{lw}$, by Newton's third law you also know $F_{wl}$.

Now repeat the analysis for the elevator alone. You have three external forces this time. One from gravity, one from the woman's feet on the lift floor and one from tension. You want to determine the one from tension. But you know the other two and the resulting acceleration. So you apply Newton's second law and solve for $F_T$. At no point does this approach require you to to add up the masses of the lift and of the woman.

[You do end up adding the gravitational forces on each, so your mathematical legerdemain has not saved any work].

Thank you so much. I understand now and can now solve as either the lift and woman together or the lift alone (after calculating normal reaction of lift on woman). Much appreciated.