Introduction to Elevator Forces during Deceleration

When a person is carrying a heavy box in an elevator that is slowing down, they often feel an interesting shift in the forces at play. This phenomenon can be explained through the lens of physics, particularly Newton's laws of motion. In this article, we will delve into the mathematical and physical principles that govern the forces of deceleration in elevators, and how they interact with objects and people inside the elevator.

Understanding Deceleration and Newton's Laws

To analyze the forces involved when an elevator is decelerating, we will employ one of the most fundamental principles of physics: Newton's third law of motion, which states that for every action, there is an equal and opposite reaction. Additionally, we will utilize Newton's second law, which states that the acceleration of an object is directly proportional to the force acting on it and inversely proportional to its mass. By applying these laws, we can calculate the forces exerted on the person and the box they are carrying.

Weight of the Box and Gravitational Force

The gravitational force acting on the box can be calculated using the formula: $$ F_{g} m cdot g $$ where ${m}$ is the mass of the box (25 kg) and ${g}$ is the acceleration due to gravity (approximately 9.81 m/s2). Substituting these values, we get: $$ F_{g} 25 text{ kg} cdot 9.81 text{ m/s}^2 245.25 text{ N} $$ This gravitational force acts downwards on the box.

Deceleration and Net Force

During deceleration, the elevator experiences an upward net force. This force is given by the formula: $$ F_{text{net}} m cdot a $$ where ${a}$ is the deceleration of the elevator (3 m/s2). Substituting the values, we get: $$ F_{text{net}} 25 text{ kg} cdot (-3 text{ m/s}^2) -75 text{ N} $$ The negative value indicates that the force is acting upwards.

Total Force Exerted by the Elevator

Next, we need to calculate the total force exerted by the elevator on the person's feet. According to Newton's second law, the force ${N}$ exerted by the elevator must balance both the gravitational force and the net force due to deceleration. Thus, we can write: $$ N - F_{g} F_{text{net}} $$ Substituting the values, we get: $$ N - 245.25 text{ N} -75 text{ N} $$ Solving for ${N}$, we find: $$ N -75 text{ N} 245.25 text{ N} 170.25 text{ N} $$ The force exerted by the elevator on the person's feet is 170.25 N.

Revisiting the Concepts with a Different Approach

Let's revisit the problem using a different perspective. By defining "down" as the negative direction and "up" as the positive direction with respect to Earth's surface, and letting gravitational acceleration ${g}$ be -9.8 m/s2, we can describe the elevator's deceleration as ${a -3 text{ m/s}^2}$. Therefore, the acceleration of the object (the person and the box) becomes ${a' a - g -3 text{ m/s}^2 - (-9.8 text{ m/s}^2) 6.8 text{ m/s}^2}$.

According to Newton's second law, the net force ${F}$ acting on the object is given by:

$$ F m cdot a' $$

Where ${m}$ is the total mass (25 kg [mass of the person] kg). Substituting the values, we get:

$$ F 25 text{ kg} cdot 6.8 text{ m/s}^2 6.8M $$

This force ${F}$ is the force exerted by the elevator floor on the object, and it is acting upwards.

Conclusion

Based on the analysis, the force exerted by the elevator on the person's feet is greater than the force exerted by the person's feet on the floor during deceleration. This is due to the deceleration effect on the elevator, which creates an upward net force.

Key Takeaways:

The force exerted by the elevator on the person's feet is 170.25 N. The force exerted by the person's feet on the floor is less than this value. The gravitational force and deceleration of the elevator affect the net force on the object.

Further Reading:

Newton's Laws of Motion Forces in Physics: Understanding Elevator Dynamics Application of Physics in Everyday Situations: The Elevator Scenario