A person weighing 800 N rides in an elevator that has a downward acceleration of 1.5 m/s2. What is the magnitude of the force of the floor of the elevator on the person?

To solve this problem, we need to find the force that the floor of the elevator exerts on the person (the normal force). We'll break this down step by step:

### Given dataTo solve this problem, we need to find the force that the floor of the elevator exerts on the person (the normal force). We'll break this down step by step:

### Given 
- Weight of the person: \( W = 800 \, \text{N} \)
- Acceleration of the elevator: \( a = 1.5 \, \text{m/s}^2 \) (downward)
- Gravitational acceleration: \( g = 9.8 \, \text{m/s}^2 \)

### Step 1: Calculate the mass of the person
The weight of a person is related to their mass through the equation \( W = mg \), where:
- \( m \) is the mass
- \( g \) is the acceleration due to gravity

So, the mass of the person is:
\[
m = \frac{W}{g} = \frac{800 \, \text{N}}{9.8 \, \text{m/s}^2} = 81.63 \, \text{kg}
\]

### Step 2: Analyze the forces on the person
The two main forces acting on the person are:
1. The gravitational force (weight), which is \( W = 800 \, \text{N} \), acting downward.
2. The normal force \( F_N \), exerted by the floor of the elevator on the person, which we are trying to find.

Since the elevator is accelerating downward, the net force is less than the gravitational force. Using Newton's Second Law, the net force \( F_{\text{net}} \) is:
\[
F_{\text{net}} = m \cdot a
\]
where \( a \) is the downward acceleration of the elevator.

### Step 3: Apply Newton’s Second Law
The net force is also given by the difference between the gravitational force and the normal force. Since the elevator is accelerating downward, we have:
\[
F_{\text{net}} = W - F_N
\]
Substituting \( F_{\text{net}} = m \cdot a \) and solving for \( F_N \):
\[
m \cdot a = W - F_N
\]
\[
F_N = W - m \cdot a
\]

### Step 4: Calculate the normal force
Now we can substitute the values:
\[
F_N = 800 \, \text{N} - (81.63 \, \text{kg}) \cdot (1.5 \, \text{m/s}^2)
\]
\[
F_N = 800 \, \text{N} - 122.45 \, \text{N}
\]
\[
F_N = 677.55 \, \text{N}
\]

### Final Answer:
The magnitude of the force that the floor of the elevator exerts on the person is approximately **677.6 N**.

Would you like further details or have any questions?

Here are 5 related questions:
1. What would the force be if the elevator was accelerating upward instead of downward?
2. How does the normal force change if the elevator moves at constant velocity?
3. What would happen if the elevator were in free fall?
4. How does the normal force compare to the person's weight when the elevator is at rest?
5. Can this concept be applied to other vehicles, such as a car accelerating on a road?

**Tip:** In problems involving elevators, the direction of acceleration is key in determining whether the normal force increases or decreases.

A person weighing 800 N rides in an elevator that has a downward acceleration of 1.5 m/s^2. What is the magnitude of the force of the floor of the elevator on the person?

Calculate the normal force acting on a person weighing 800 N in a downward-accelerating elevator at 1.5 m/s^2. This solution involves applying Newton's Second Law and understanding the relationship between forces and acceleration, suitable for Grades 9-12.

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