Forces exerted on a body inside an accelerating elevator

Question

Three masses $m_1 = 3\text{ kg}$, $m_2 = 9\text{ kg}$ and $m_3 = 6\text{ kg}$ hang from three identical springs in a motionless elevator. The elevator is moving downward with a velocity of $v = -2\text{ m/s}$ but accelerating upward with an acceleration of $a = 5\ \mathrm{m/s^2}$. (Note: an upward acceleration when the elevator is moving down means the elevator is slowing down.) What is the force the bottom spring exerts on the bottom mass? Take $g = 10\ \mathrm{m/s^2}$.

My argument

The elevator is steadily slowing down and it get an upward acceleration.

So as it goes down the net force is

$$\begin{gather}F_s - mg = ma \\ F_s = 6(10 + 5) = 6(15) = 90\text{ N}\end{gather}$$

But this is the force the spring has. It is the same as the force EXERTED on the body attached? Is there a Newton's third law involved?

Friend's argument

He says that BECAUSE it has an upward acceleration and it is going down that we need to be concern with

$$Fs = mg - ma = 90 - 60 = 30\text{ N}$$

Answer 1

As Mark wrote, your reasoning is (almost) correct and your friend's is not. Here's the sort of explanation I use when I'm teaching this topic:

In order to apply Newton's second law, you need to choose one object to apply it to. Ignore everything else except the object you choose and the forces that act on it. In your case, the problem asks about a force exerted on the bottom mass, so you should write out Newton's second law for the bottom mass.

There are exactly three quantities that go into the equation:

1. Mass of the object
   This is given in the problem. (Typically the mass is given.)
2. Acceleration of the object
   This is also given in the problem. (In most cases, it is either given or it is what you will be solving for.) When you are determining acceleration, only consider how the object is changing its velocity. Gravity is irrelevant, the value and direction of velocity are irrelevant, whether the object is slowing down or speeding up or curving is irrelevant. The problem tells you that the elevator is accelerating at $5\ \mathrm{m/s^2}$ upward, so the only thing you need to assume is that the mass is moving along with the elevator, and that means the acceleration of the mass is $a = 5\ \mathrm{m/s^2}$ (assuming positive values are upward).

3. Net force on the object
   This is usually the part of Newton's law that takes a bit of thought. You have to enumerate each of the forces acting on the object and include the correct term for each one. In this case, there are two forces acting on the object (which is the lowest mass): the force exerted by the spring above, and gravity. The spring force goes up and gravity goes down, so you would write the net force as $F_s - F_g$, or $F_s - mg$ once you plug in the value of the gravitational force.

   Note that each of these terms represents a force on the object. $F_s$ is the force that the spring exerts on the mass; $F_g$ is the force that the Earth exerts on the mass. The problem asks for the force that the spring exerts on the mass, so you do not need to invoke Newton's third law in this case.

Once you have all these quantities, you can put them into the equation

$$\sum F = ma$$

and solve for whatever you need to.

Answer 2

If we assume the elevator's motion is adiabatic, so that the springs are never set oscillating, then your answer is correct.

The velocity of the elevator is irrelevant due to the principle of relativity.

The equivalence principle states that when the elevator accelerates, the effects are indistinguishable from those of a gravitational field. Thus, when the elevator accelerates up at $5 m/s$, regardless of its speed, the physics is the same as if the elevator were stationary in a gravitational field whose acceleration is $15 m/s$.

If the elevator's acceleration changes on time scales similar to the damping time of the springs, the masses will oscillate and the force will not be determined from the given information.

Answer 3

According to Newton's 2nd law, net force acting on the object is mass times acceleration. Here, as we see from the free body diagram of the lowest mass, there are two forces:

1. the weight $m \cdot g$
2. the force spring exerts on the mass.

Hence as force is a vector quantity and so is acceleration so we have to write 2nd law as

$F - m \cdot g = m \cdot a$ and hence $F = m \cdot g + m \cdot a$