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, we need to consider all the forces acting on the person and the elevator.
1. Weight (W): The weight of the person acting downwards (691 N).
2. Normal force (N): The normal force exerted by the scale, acting upwards (499 N).
3. Tension force (T): The tension force acting upwards due to the elevator accelerating (unknown).
--------------
| | Person
| | Weight (W = 691 N)
| |
| Scale | Normal force (N = 499 N)
| |
| |
|_______________| Elevator
(b) The qualitative vertical equation for this situation can be described using Newton's second law, which states that the net force acting on an object is equal to the mass of the object multiplied by its acceleration.
For the person standing in the elevator:
Sum of forces vertically = mass x acceleration
Weight (W) - Normal force (N) = mass x acceleration
(c) To calculate the magnitude and direction of the elevator's acceleration, we can use the qualitative vertical equation above and solve for acceleration.
691 N - 499 N = mass x acceleration
Acceleration = (691 N - 499 N) / mass
To find the mass, we can use the equation Weight = mass x gravity.
Weight = 691 N
Gravity (g) = 9.8 m/s^2 (acceleration due to gravity)
mass = Weight / gravity
mass = 691 N / 9.8 m/s^2
Once we have the mass, we can substitute it back into the previous equation and solve for acceleration. The direction of the acceleration will depend on the direction of the force causing it. If the elevator is moving upwards, the acceleration will be upwards. If the elevator is moving downwards, the acceleration will be downwards.
(a) To draw a force diagram of the situation, we need to consider all the forces acting on the person in the elevator.
First, there is the gravitational force acting vertically downward, with a magnitude of 691 N. This force is the person's weight.
Second, there is the normal force exerted by the scale in the elevator. When the person stands on the scale, this force acts vertically upward and has a magnitude equal to the reading on the scale, which is 499 N in this case. This force is exerted by the scale to counteract the weight of the person.
So, the force diagram will have two forces, one pointing downward and one pointing upward, as shown below:
Person: ---> 691 N downward (Weight)
<--- 499 N upward (Normal force)
(b) To create a qualitative vertical equation, we can use Newton's second law, which states that the net force acting on an object is equal to the object's mass multiplied by its acceleration.
In this situation, the only forces acting on the person in the vertical direction are the weight (691 N downward) and the normal force (499 N upward). Since the person is standing on the scale in a motionless elevator, the net force in the vertical direction is zero. Therefore, we can create the following equation:
Net vertical force = Weight - Normal force = 0
(c) To calculate the magnitude and direction of the elevator's acceleration, we need to consider the net force acting on the person, which is the difference between the weight and the normal force.
Net vertical force = Weight - Normal force = 0
Rearranging the equation, we get:
Weight = Normal force
Since the weight is equal to 691 N and the normal force is equal to 499 N, this equation is not satisfied. This means the net vertical force is not zero, indicating that there is an acceleration in the vertical direction.
To calculate the magnitude of the elevator's acceleration, we can use Newton's second law. We know that the net force is the mass of the person multiplied by the acceleration:
Net vertical force = mass x acceleration
Substituting the weight for the net vertical force:
Weight = mass x acceleration
691 N = mass x acceleration
Since the mass is not provided in the question, we cannot calculate the exact magnitude of the elevator's acceleration without this information. However, we can see that the acceleration will be in the upward direction since the normal force is less than the weight.