A woman stands on a scale in a moving elevator. Her mass is 60.0 kg, and the combined mass of the elevator and scale is an additional 815 kg. Starting from rest, the elevator accelerates upward. During the acceleration, the hoisting cable applies a force of 9410 N. What does the scale read during the acceleration?
acceleration= force/(total mass)
now, knowing acceleration
weight= masswoman(g+a)
To find what the scale reads during the acceleration, we need to consider the forces acting on the woman in the elevator.
1. Weight force (mg): The weight of the woman is given by the mass multiplied by the acceleration due to gravity (g ≈ 9.8 m/s²). So, the weight force is given by F_weight = mg.
2. Normal force (N): Since the woman is standing on the scale, the scale exerts an equal and opposite normal force on her. This normal force cancels out a part of the weight force, which is why the scale reading is lower than the actual weight. The normal force can be calculated using Newton's second law in the vertical direction: ∑ F_y = ma_y.
3. Tension force (T): The hoisting cable applies a force to the elevator to accelerate it upward. This force is equal to the sum of the weight of the elevator and scale (combined mass) and the tension force, so T = F_weight + F_elevator.
Let's calculate the scale reading step by step:
1. Calculate the weight force:
F_weight = mg = 60.0 kg × 9.8 m/s² = 588 N
2. Calculate the tension force:
T = F_weight + F_elevator
= 588 N + 9410 N
= 9998 N
3. Calculate the normal force:
Since the elevator is accelerating upwards, the net force in the vertical direction is upward.
∑ F_y = ma_y
N - F_weight = ma
N = ma + F_weight
We know that the acceleration of the elevator is the same as the acceleration experienced by the woman, so a = 9410 N / (60.0 kg + 815 kg) = 11.07 m/s²
N = (60.0 kg + 815 kg) × 11.07 m/s² + 588 N
N = 8980.5 N + 588 N
N = 9568.5 N
Therefore, the scale reading during the acceleration is 9568.5 Newtons.
To determine what the scale reads during the acceleration, we need to consider the forces acting on the woman.
1. First, let's calculate the total force acting on the woman. We know that force is equal to mass multiplied by acceleration:
Force = mass * acceleration
The force acting on the woman is the sum of her weight (mg) and the force exerted by the scale. However, since the elevator is accelerating, there will also be a pseudo force acting on her. The pseudo force is equal to the mass of the woman multiplied by the acceleration of the elevator:
Total Force = weight + force by the scale + pseudo force
Using this equation, we can calculate the total force acting on the woman:
Total Force = (mass * acceleration) + (mass * g) + (mass * acceleration)
2. Next, we need to find the weight of the woman, which is equal to her mass multiplied by the acceleration due to gravity (g). On Earth, the acceleration due to gravity is approximately 9.8 m/s^2.
Weight = mass * g
Weight = 60.0 kg * 9.8 m/s^2
3. We also need to calculate the pseudo force, which is equal to the mass of the woman multiplied by the acceleration of the elevator:
Pseudo Force = mass * acceleration
Pseudo Force = 60.0 kg * acceleration
4. Substituting these values back into the total force equation:
Total Force = (60.0 kg * acceleration) + (60.0 kg * 9.8 m/s^2) + (60.0 kg * acceleration)
Total Force = (120.0 kg * acceleration) + (588.0 N)
5. The force exerted by the scale is equal to the scale reading. Therefore, we can equate the total force to the force exerted by the scale:
Total Force = Force by the scale
(120.0 kg * acceleration) + (588.0 N) = 9410 N
6. Rearranging the equation to solve for the acceleration:
(120.0 kg * acceleration) = 9410 N - 588.0 N
(120.0 kg * acceleration) = 8822 N
acceleration = 8822 N / 120.0 kg
acceleration ≈ 73.5 m/s^2
7. Now that we know the acceleration, we can find the scale reading using the equation for the force exerted by the scale:
Force by the scale = mass * acceleration + weight
Force by the scale = 60.0 kg * 73.5 m/s^2 + (60.0 kg * 9.8 m/s^2)
Force by the scale ≈ 4410 N + 588.0 N
Force by the scale ≈ 4998 N
Therefore, the scale will read approximately 4998 N during the acceleration of the elevator.