Question

A woman standing on a scale in a moving elevator, mass 65.0 kg, elevator and scale mass 757 kg, hoisting cable force 9550 N.

Answer 1

To analyze the situation described, we first identify the forces acting on the woman, the elevator, and the scale.

### Given Data:
- Mass of the woman, \( m_w = 65.0 \, \text{kg} \)
- Mass of the elevator and scale, \( m_{e} = 757 \, \text{kg} \)
- Force in the hoisting cable, \( F_c = 9550 \, \text{N} \)

### Force Analysis:
1. **Weight of the woman:**
\[
W_w = m_w \cdot g = 65.0 \, \text{kg} \cdot 9.81 \, \text{m/s}^2 = 637.65 \, \text{N}
\]

2. **Weight of the elevator and scale:**
\[
W_e = m_e \cdot g = 757 \, \text{kg} \cdot 9.81 \, \text{m/s}^2 = 7413.57 \, \text{N}
\]

3. **Total weight (combined) when the elevator is at rest:**
\[
W_{total} = W_w + W_e = 637.65 \, \text{N} + 7413.57 \, \text{N} = 8051.22 \, \text{N}
\]

### Net Force and Acceleration:
The net force acting on the elevator system (including the woman) can be determined using:
\[
F_{net} = F_c - W_{total}
\]
To find the acceleration of the elevator \( a \):
\[
F_{net} = (m_w + m_e) \cdot a
\]

Substituting the known values:
\[
F_c - W_{total} = (m_w + m_e) \cdot a
\]
\[
9550 \, \text{N} - 8051.22 \, \text{N} = (65.0 \, \text{kg} + 757 \, \text{kg}) \cdot a
\]
\[
1498.78 \, \text{N} = 822 \, \text{kg} \cdot a
\]

Now, solving for \( a \):
\[
a = \frac{1498.78 \, \text{N}}{822 \, \text{kg}} \approx 1.82 \, \text{m/s}^2
\]

### Apparent Weight on the Scale:
The apparent weight (the reading on the scale) can be calculated using:
\[
F_{scale} = W_w + m_w \cdot a
\]
\[
F_{scale} = 637.65 \, \text{N} + 65.0 \, \text{kg} \cdot 1.82 \, \text{m/s}^2
\]
\[
F_{scale} = 637.65 \, \text{N} + 118.3 \, \text{N} \approx 755.95 \, \text{N}
\]

### Conclusion:
- The acceleration of the elevator is approximately \( 1.82 \, \text{m/s}^2 \).
- The scale reading (apparent weight of the woman) is approximately \( 755.95 \, \text{N} \).