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.
positive is force up on feet
691 up - m g down = 0 if motionless
if g = 9.81, m = 70.44 kg
F - m g = m a
where a is up
499 - 691 = 70.44 a
a = -2.73
so accelerating down (negative a)
(a) To draw a force diagram of the situation, let's consider all the forces acting on the person in the elevator.
First, we have the weight of the person, which is a force of 691 N directed downwards. This force is the force exerted by the Earth's gravity on the person.
Second, we have the normal force exerted by the bathroom scale on the person. Initially, when the elevator is motionless, this force is equal in magnitude to the weight of the person, which is 691 N, and it acts upwards. However, when the elevator begins to move upwards, this normal force decreases. This is because when the elevator accelerates upwards, the person experiences a pseudo-force in the opposite direction, making the apparent weight decrease. The scale reads a value of 499 N, indicating that the normal force has decreased.
The force diagram would look like this:
--------- X --------- (Weight of the person, 691 N, downwards)
^
Normal force (initially 691 N, upwards, but decreases when the elevator moves)
(b) To create a qualitative vertical equation for this situation, we can consider the forces in the vertical direction.
Initially, when the elevator is motionless, the net force in the vertical direction is zero since the person is at rest. This means that the normal force is equal in magnitude to the weight, which can be represented as:
Normal force = Weight of the person (691 N)
When the elevator begins to move upwards, the net force in the vertical direction is not zero. The normal force decreases, resulting in a difference between the weight of the person and the normal force. This difference is equal to the force exerted due to the acceleration of the elevator.
The equation can be represented as:
Weight of the person - Normal force = Force due to acceleration
(c) To calculate the magnitude and direction of the elevator's acceleration, we use the equation from part (b):
Force due to acceleration = Weight of the person - Normal force
Force due to acceleration = 691 N - 499 N
Force due to acceleration = 192 N (upwards)
The magnitude of the elevator's acceleration is 192 N, and it is directed upwards.