While in an elevator, you decide to take some weight readings of yourself to determine the elevator’s acceleration. When the elevator is stopped, you read a weight of 59.2 kg. When the elevator is moving, you measure a maximum weight of 63.5 kg, and a minimum weight of 55.9 kg.
(a)
What is the maximum upward acceleration of the elevator?
(b)
What is the maximum downward acceleration of the elevator?
To determine the maximum upward and downward accelerations of the elevator, we can use the concept of apparent weight.
(a) To find the maximum upward acceleration of the elevator, we can use the maximum weight reading while the elevator is moving.
First, we need to calculate the normal force when the elevator is moving. The normal force is equal to the weight when there is no acceleration. Therefore, the normal force is equal to the weight reading when the elevator is stopped, which is 59.2 kg.
Next, we need to calculate the apparent weight when the elevator is moving. The apparent weight is the weight reading minus the force due to the acceleration. So, the apparent weight is 63.5 kg - the force due to the upward acceleration.
Using Newton's second law, we know that the net force acting on an object is equal to its mass multiplied by its acceleration. In this case, the net force is the difference between the normal force and the apparent weight. So, we have:
Net force = Normal force - Apparent weight
= m * g - m * a
Where m is the mass, g is the acceleration due to gravity (9.8 m/s^2), and a is the upward acceleration of the elevator.
Setting this net force equal to mass multiplied by acceleration:
m * g - m * a = m * a
Simplifying the equation:
m * g = 2 * m * a
a = g / 2
The maximum upward acceleration of the elevator is equal to half of the acceleration due to gravity, which is approximately 4.9 m/s^2.
(b) To find the maximum downward acceleration of the elevator, we can use the minimum weight reading while the elevator is moving.
Following the same steps as in part (a), we can calculate the apparent weight when the elevator is moving downward. The apparent weight is the weight reading minus the force due to the acceleration. So, the apparent weight is 55.9 kg + the force due to the downward acceleration.
Similarly, the normal force is still equal to 59.2 kg.
Using Newton's second law as before:
Net force = Normal force - Apparent weight
= m * g - m * a
Setting this net force equal to mass multiplied by acceleration:
m * g - m * a = -m * a
Simplifying the equation:
m * g = 0
a = 0
The maximum downward acceleration of the elevator is zero, because when the elevator is moving downward, there is no net force acting on it.