A person with the weight of 691 N stands on a bathroom scale in a motionless elevator. The elevator begins to move and the scale momentarily changes to 499 N. (a) Draw a force diagram of the situation. (b) Create a qualitative vertical equation for this situation. (c) Calculate the magnitude and direction of the elevator’s acceleration.

(a) To draw a force diagram of the situation, you need to consider all the forces acting on the person in the elevator.

First, there is the gravitational force acting on the person, which can be represented as a downward arrow labeled "W" for weight, with a magnitude of 691 N.

Second, there is the normal force exerted by the bathroom scale on the person, which opposes the gravitational force and cancels it out when the person is at rest. This force can be represented as an upward arrow labeled "N" with a magnitude of 691 N.

Third, when the elevator starts to move upward, there is an additional force acting on the person in the downward direction, which can be labeled as "F" for force due to acceleration. This force is responsible for the change in reading on the scale.

(b) To create a qualitative vertical equation for this situation, you need to consider the forces acting in the vertical direction.

When the person is at rest, the normal force ("N") exerted by the bathroom scale cancels out the weight ("W") of the person. So, N = W.

However, when the elevator starts to move, the normal force is decreased due to the force due to acceleration ("F") acting in the downward direction. Therefore, the equation can be written as N - F = W.

(c) To calculate the magnitude and direction of the elevator's acceleration, you can apply Newton's second law of motion, which states that the net force acting on an object is equal to its mass multiplied by its acceleration (F = ma).

In this case, the net force acting on the person is the difference between the normal force ("N") and the weight ("W"), which is equal to the force due to acceleration ("F"). So, F = N - W.

Substituting the given values, F = 499 N - 691 N = -192 N.

Since Force = mass * acceleration, we can rearrange the equation to find the acceleration (a) by dividing both sides by the mass of the person.

Therefore, a = F / m.

The magnitude of the acceleration is given by |a| = |F| / m = |-192 N| / m.

The direction of the acceleration is downwards because the force due to acceleration is acting in the downward direction, which matches the direction of the decrease in the normal force.

Note that to calculate the acceleration, you would also need to know the mass (m) of the person, which is not provided in the question.

(a) Here is a force diagram for the situation:

upward force from the scale (499 N)
| ↓
| ↑
| 691 N (weight)
| ↓
| ↑
--------------------------------
← Floor of the elevator →

(b) The qualitative vertical equation for this situation is:
Net force = mass × acceleration

(c) To find the magnitude and direction of the elevator's acceleration, we need to first calculate the net force. The net force is the difference between the force from the scale (499 N) and the weight (691 N).

Net force = 499 N - 691 N
Net force = -192 N

Since the net force is negative, we know that the elevator is accelerating downward.

Next, we can use the equation Net force = mass × acceleration to solve for the acceleration. The mass is not given in the problem, but we can use the weight of the person to find it.

Weight = mass × gravity

Rearranging the equation, we have:

mass = weight / gravity

mass = 691 N / 9.8 m/s²
mass = 70.51 kg

Now we can plug in the values into the equation to solve for acceleration:

-192 N = 70.51 kg × acceleration

Rearranging the equation to solve for acceleration:

acceleration = -192 N / 70.51 kg
acceleration = -2.72 m/s²

So, the magnitude of the elevator's acceleration is 2.72 m/s², and it is directed downward.