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 ?

Options

A. 588 N
B. 606 N
C. 645 N
D. 720 N

Answer

To solve the problem, we need to determine the reading on the scale when the elevator is accelerating upward. We will follow these steps:

### Step 1: Identify the forces acting on the system
The forces acting on the woman in the elevator include:
1. The gravitational force (weight) acting downward.
2. The normal force (the reading on the scale) acting upward.

### Step 2: Calculate the weight of the woman
The weight \( W \) of the woman can be calculated using the formula:
\[ W = m \cdot g \]
where:
- \( m = 60.0 \, \text{kg} \) (mass of the woman)
- \( g = 9.8 \, \text{m/s}^2 \) (acceleration due to gravity)

Calculating the weight:
\[ W = 60.0 \, \text{kg} \cdot 9.8 \, \text{m/s}^2 = 588 \, \text{N} \]

### Step 3: Determine the total mass of the elevator system
The total mass \( M \) of the elevator system (including the scale and the woman) is:
\[ M = m_{\text{woman}} + m_{\text{elevator and scale}} \]
where:
- \( m_{\text{elevator and scale}} = 815 \, \text{kg} \)

Calculating the total mass:
\[ M = 60.0 \, \text{kg} + 815 \, \text{kg} = 875 \, \text{kg} \]

### Step 4: Calculate the net force acting on the elevator system
The tension \( T \) in the cable is given as \( 9410 \, \text{N} \). The net force \( F_{\text{net}} \) acting on the system can be calculated using Newton's second law:
\[ F_{\text{net}} = T - W_{\text{total}} \]
where \( W_{\text{total}} \) is the total weight of the elevator system.

Calculating the total weight:
\[ W_{\text{total}} = M \cdot g = 875 \, \text{kg} \cdot 9.8 \, \text{m/s}^2 = 8585 \, \text{N} \]

Now, substituting the values:
\[ F_{\text{net}} = 9410 \, \text{N} - 8585 \, \text{N} = 825 \, \text{N} \]

### Step 5: Calculate the acceleration of the elevator
Using Newton's second law, the acceleration \( a \) can be calculated as:
\[ F_{\text{net}} = M \cdot a \]
Thus,
\[ a = \frac{F_{\text{net}}}{M} = \frac{825 \, \text{N}}{875 \, \text{kg}} \approx 0.943 \, \text{m/s}^2 \]

### Step 6: Calculate the apparent weight of the woman
The apparent weight (the reading on the scale) when the elevator is accelerating upward can be calculated using:
\[ F_{\text{apparent}} = W + m \cdot a \]
Substituting the values:
\[ F_{\text{apparent}} = 588 \, \text{N} + (60.0 \, \text{kg} \cdot 0.943 \, \text{m/s}^2) \]
\[ F_{\text{apparent}} = 588 \, \text{N} + 56.58 \, \text{N} \approx 644.58 \, \text{N} \]

### Step 7: Conclusion
The scale reads approximately \( 644.58 \, \text{N} \), which can be rounded to \( 645 \, \text{N} \).