A woman stands on a scale in a moving elevator. Her mass is 57.0 kg, and the combined mass of the elevator and scale is an additional 811 kg. Starting from rest, the elevator accelerates upward. During the acceleration, the hoisting cable applies a force of 9391 N. What does the scale read during the acceleration?
To find out what the scale reads during the acceleration, we need to consider the forces acting on the woman and the elevator.
First, let's calculate the net force acting on the woman-elevator system during the acceleration.
The force applied by the hoisting cable, Fcable = 9391 N (upward)
The weight of the woman, Fweight_woman = mg = 57.0 kg * 9.8 m/s^2 ≈ 558.6 N (downward)
The weight of the elevator and scale, Fweight_elevator = mg = 811 kg * 9.8 m/s^2 ≈ 7941.8 N (downward)
Next, let's calculate the net force.
Net force = Fcable + Fweight_woman + Fweight_elevator
= 9391 N + (-558.6 N) + (-7941.8 N) = 887.6 N upward
Now, let's calculate the apparent weight of the woman, which is equal to the reading on the scale.
Apparent weight = Fweight_woman + Fweight_elevator
= -558.6 N + (-7941.8 N) = -8500.4 N
Since the scale reads the normal force, which is equal to the apparent weight, the scale reads approximately -8500.4 N during the acceleration.
Note: The negative sign indicates that the scale reads a value in the opposite direction of the gravitational force. In this case, the scale reading would be negative, indicating that the woman feels a reduced apparent weight during the upward acceleration.
To determine what the scale reads during the acceleration, we need to calculate the net force acting on the woman. This net force can be calculated using Newton's second law of motion, which states that the net force is equal to the mass multiplied by the acceleration.
Given information:
- Mass of the woman = 57.0 kg
- Combined mass of the elevator and scale = 811 kg
- Force applied by the hoisting cable = 9391 N
First, let's calculate the total mass of the system (woman + elevator + scale):
Total mass = Mass of the woman + Mass of the elevator and scale
Total mass = 57.0 kg + 811 kg
Total mass = 868.0 kg
Now, let's calculate the net force acting on the system:
Net force = Force applied by hoisting cable - Weight of the system
Weight of the system = Total mass * acceleration due to gravity
The weight of an object is equal to its mass multiplied by the acceleration due to gravity (approximately 9.8 m/s^2).
Weight of the system = 868.0 kg * 9.8 m/s^2
Weight of the system = 8514.4 N
Net force = 9391 N - 8514.4 N
Net force = 876.6 N
Now that we have the net force on the system, we can calculate the acceleration:
Net force = Total mass * acceleration
876.6 N = 868.0 kg * acceleration
Simplifying:
acceleration = 876.6 N / 868.0 kg
acceleration = 1.010 m/s^2
Finally, we can determine what the scale reads during the acceleration. The scale reading is equal to the normal force acting on the woman, which is equal to her weight.
Weight of the woman = mass of the woman * acceleration due to gravity
Weight of the woman = 57.0 kg * 9.8 m/s^2
Weight of the woman = 558.6 N
Therefore, the scale reads 558.6 Newtons during the acceleration.