A woman stands on a scale in a moving elevator. Her mass is 56.0 kg, and the combined mass of the elevator and scale is an additional 809 kg. Starting from rest, the elevator accelerates upward. During the acceleration, the hoisting cable applies a force of 9030 N. What does the scale read during the acceleration?
To determine what the scale reads during the acceleration, we need to consider the forces acting on the woman when she is in the elevator.
There are three forces acting on the woman:
1. Gravitational force (weight) pulling her downward, given by: F_gravity = mass * acceleration due to gravity
2. Normal force exerted by the scale pushing her upward, which is what the scale reads.
3. Force applied by the hoisting cable pushing her upward, since the elevator is accelerating.
Let's break down the problem and solve it step by step:
Step 1: Determine the gravitational force.
The gravitational force acting on the woman can be calculated using the equation: F_gravity = mass * acceleration due to gravity.
Given that her mass is 56.0 kg and the acceleration due to gravity is approximately 9.8 m/s², we can calculate the gravitational force:
F_gravity = 56.0 kg * 9.8 m/s² = 549.6 N (approximately)
Step 2: Calculate the net force acting on the woman.
To determine the net force, we need to consider the forces acting in the vertical direction. Since the elevator is accelerating upward, the net force will be the difference between the forces pushing upward and those pushing downward.
Net force = Force applied by hoisting cable - Gravitational force.
Given that the force applied by the hoisting cable is 9030 N, we can calculate the net force:
Net force = 9030 N - 549.6 N = 8480.4 N (approximately)
Step 3: Determine the scale reading.
The scale reading is equivalent to the normal force exerted by the scale on the woman. In this case, the scale reading is also equal to the net force calculated in Step 2. Therefore, the scale reading during the acceleration is approximately 8480.4 N.
So, the scale reading during the upward acceleration of the elevator is approximately 8480.4 N.
To find out 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. This force is equal to the force exerted by the scale (which the scale will read) plus the force due to gravity acting on the woman.
The force due to gravity (weight) can be calculated using the formula:
Weight = mass * gravity, where the mass is 56.0 kg and gravity is approximately 9.8 m/s^2.
Weight = 56.0 kg * 9.8 m/s^2 = 548.8 N
Therefore, the total force acting on the woman is the sum of the force due to gravity and the force applied by the cable:
Total Force = Weight + Force Applied by Cable
Total Force = 548.8 N + 9030 N = 9578.8 N
2. Next, let's determine the net force acting on the woman. Net force is the force that causes the acceleration.
Net Force = Total Force - Force of Elevator and Scale
Net Force = 9578.8 N - 809 kg * 9.8 m/s^2 ≈ 9578.8 N - 7932 N
Net Force = 1646.8 N
3. Finally, we can calculate the scale reading. The scale reading is equal to the magnitude of the normal force exerted by the scale on the woman.
According to Newton's second law, the net force is equal to the mass times the acceleration:
Net Force = Mass * Acceleration
Solving for the acceleration:
Acceleration = Net Force / Mass
Acceleration = 1646.8 N / 56.0 kg ≈ 29.44 m/s^2
Since the scale reading is the normal force, which is acting in the upward direction, it can be calculated using the formula:
Scale Reading = Mass * (Gravity + Acceleration)
Scale Reading = 56.0 kg * (9.8 m/s^2 + 29.44 m/s^2) ≈ 2457 N
Therefore, the scale will read approximately 2457 N during the acceleration.