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.
m = 691/9.8
(c) F = ma
R=net force, And net force =mg-ma,m(g-a). R=499,m=691/9.8=70.51kg 499=70.51(9.8-a),find a,and a is directed upward
To find the magnitude and direction of the elevator's acceleration in this situation, we can follow these steps:
(a) Draw a force diagram of the situation:
In this scenario, we have a person standing on a bathroom scale in a motionless elevator. We need to consider two forces acting on the person: the person's weight (691 N) acting downwards and the normal force exerted by the scale, which is equal to the reading on the scale (499 N) and acts upwards. These forces can be represented on a force diagram as follows:
+--------+
| normal |
+--------+
| person |
+--------+
| |
<<<<<<<<
(b) Create a qualitative vertical equation for this situation:
In the vertical direction, the weight of the person (downwards) and the normal force (upwards) must balance each other for the person to be in equilibrium (not accelerating):
Weight = Normal Force
This equation states that the magnitude of the weight of the person must be equal to the magnitude of the normal force exerted by the scale.
(c) Calculate the magnitude and direction of the elevator's acceleration:
When the elevator starts to move, the reading on the scale momentarily decreases to 499 N. This decrease in the normal force indicates that there is a net force acting downwards, causing the person's apparent weight to be less than their actual weight. Therefore, there must be an unbalanced force in the downward direction, which is the net force.
To calculate the magnitude of the net force, we can subtract the normal force from the weight:
Net Force = Weight - Normal Force
= 691 N - 499 N
= 192 N
Since the net force is acting downwards, this tells us that there must be an acceleration in the downward direction. To find the magnitude of the acceleration, we can use Newton's second law:
Net Force = Mass * Acceleration
We need to convert the weight from Newtons (N) to kilograms (kg) to use it in the equation. The weight is equal to the mass multiplied by the acceleration due to gravity (approximately 9.8 m/s²):
Weight = Mass * Acceleration due to gravity
691 N = Mass * 9.8 m/s²
Solving for mass:
Mass = 691 N / 9.8 m/s² ≈ 70.4 kg
Now we can rearrange Newton's second law to solve for acceleration:
Acceleration = Net Force / Mass
Acceleration = 192 N / 70.4 kg ≈ 2.73 m/s²
The magnitude of the elevator's acceleration is approximately 2.73 m/s². Since the net force is acting downwards, the direction of the elevator's acceleration is also downwards.