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.
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(a) To draw a force diagram of the situation, we need to consider the forces acting on the person in the elevator. We have the person's weight acting downward, which we'll represent as W. We also have the normal force exerted by the scale, which we'll label as N. Since the elevator is in motion, there will be an additional force due to acceleration, which we'll denote as F. The force of friction can be neglected since it's motionless. So the force diagram would look like this:
↓ W
--------------
↑ N
--------------
↑ F
(b) Creating a qualitative vertical equation involves considering the forces acting in the vertical direction. In this case, we have the weight of the person, W, acting downward and the normal force, N, acting upward. The equation would be:
Net force = W - N
(c) To calculate the magnitude and direction of the elevator's acceleration, we'll use the equation:
Net force = mass × acceleration
Since the person's weight is equal to the mass of the person multiplied by the acceleration due to gravity (W = mg), we can rewrite the equation as:
mg - N = m × a
Since the person's weight is given as 691 N and the scale reading is 499 N, we can rewrite the equation as:
691 N - 499 N = m × a
192 N = m × a
To find the magnitude of the elevator's acceleration, we plug in the mass of the person. Let's assume the mass is 70 kg (you can use a different value if required). So:
192 N = 70 kg × a
Solving for a, we have:
a = 192 N / 70 kg
Calculating this gives:
a ≈ 2.74 m/s²
Since the net force is directed downward, the acceleration of the elevator is also downward.