To find the reading of the weighing machine (which measures the normal force exerted by the man on the scale), we can analyze each of the three scenarios based on the forces acting on the man.
1. Elevator Ascending with an Acceleration of 2 m/s²
When the elevator ascends with an acceleration, the apparent weight (reading on the scale) can be found using the formula:
\[ F_{N} = m(g + a) \]
Where:
- \( F_{N} \) is the normal force (reading on the scale),
- \( m \) is the mass of the man (80 kg),
- \( g \) is the acceleration due to gravity (approximately \( 9.81 , \text{m/s}^2 \)),
- \( a \) is the acceleration of the elevator (2 m/s²).
Calculating the normal force:
\[ F_{N} = 80 , \text{kg} \times (9.81 , \text{m/s}^2 + 2 , \text{m/s}^2) = 80 , \text{kg} \times 11.81 , \text{m/s}^2 \approx 944.8 , \text{N} \]
2. Elevator Moving with Uniform Velocity
When the elevator moves with a uniform velocity, the acceleration is zero. Therefore, the reading on the scale is equal to the gravitational force:
\[ F_{N} = mg \]
Calculating the normal force:
\[ F_{N} = 80 , \text{kg} \times 9.81 , \text{m/s}^2 \approx 784.8 , \text{N} \]
3. Elevator Coming to Rest with a Retardation of 4 m/s²
When the elevator is decelerating (retardation), the apparent weight can be found using the formula:
\[ F_{N} = m(g - a) \]
Where:
- \( a \) is the retardation (4 m/s²).
Calculating the normal force:
\[ F_{N} = 80 , \text{kg} \times (9.81 , \text{m/s}^2 - 4 , \text{m/s}^2) = 80 , \text{kg} \times 5.81 , \text{m/s}^2 \approx 464.8 , \text{N} \]
Summary of Readings on the Weighing Machine:
- Ascending with 2 m/s²: \( \approx 944.8 , \text{N} \)
- Uniform Velocity: \( \approx 784.8 , \text{N} \)
- Coming to Rest with 4 m/s²: \( \approx 464.8 , \text{N} \)
These readings reflect the changes in normal force experienced by the man due to the acceleration or deceleration of the elevator.
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