Solving Elevator Problems in Physics

Elevator problems are a great way to understand the concept of apparent weight and how forces interact in an accelerating system. In this post, we’ll break down elevator scenarios step by step and show you how to solve for apparent weight changes during upward and downward acceleration.

Table of Contents

Step-by-Step Recipe for Solving Elevator Problems

1. Identify all the forces acting on the person or object in the elevator.

The forces involved are:

  • Gravitational force \( F_g = m \cdot g \) (downward, always the same).
  • Normal force \( F_n \) (the force exerted by the scale or floor, this is the apparent weight).

2. Set up the problem based on the acceleration of the elevator.

If the elevator is accelerating upwards or downwards, it changes the apparent weight felt by the person or object.

3. Apply Newton’s second law to the vertical forces.

Newton’s second law is given as:

\( \sum F = m \cdot a \)

For the vertical direction, the forces acting on the person are the gravitational force and the normal force. Depending on whether the elevator is accelerating up or down, the equation changes:

  • For upward acceleration: \( F_n = m \cdot (g + a) \)
  • For downward acceleration: \( F_n = m \cdot (g - a) \)

Worked Example: Apparent Weight in an Elevator

Problem: A person with a mass of 70 kg stands on a scale inside an elevator. Calculate the apparent weight (the normal force) when:
  • (a) The elevator is stationary or moving at constant velocity.
  • (b) The elevator is accelerating upwards at \( 2 \, \text{m/s}^2 \).
  • (c) The elevator is accelerating downwards at \( 2 \, \text{m/s}^2 \).
Step 1: Identify the forces.
  • Gravitational force: \( F_g = m \cdot g = 70 \, \text{kg} \cdot 9.8 \, \text{m/s}^2 = 686 \, \text{N} \)
  • Normal force: \( F_n \) (this is what we need to solve for).
Step 2: Apply Newton’s second law.

(a) When the elevator is stationary or moving at constant velocity:

There is no net acceleration, so:

\( F_n = m \cdot g = 686 \, \text{N} \)

Thus, the apparent weight is \( 686 \, \text{N} \), which is equal to the true weight.

(b) When the elevator is accelerating upwards at \( 2 \, \text{m/s}^2 \):

We use the equation for upward acceleration:

\( F_n = m \cdot (g + a) = 70 \, \text{kg} \cdot (9.8 + 2) \, \text{m/s}^2 = 826 \, \text{N} \)

The apparent weight is \( 826 \, \text{N} \), meaning the person feels heavier during upward acceleration.

(c) When the elevator is accelerating downwards at \( 2 \, \text{m/s}^2 \):

We use the equation for downward acceleration:

\( F_n = m \cdot (g - a) = 70 \, \text{kg} \cdot (9.8 - 2) \, \text{m/s}^2 = 546 \, \text{N} \)

The apparent weight is \( 546 \, \text{N} \), meaning the person feels lighter during downward acceleration.

Key Takeaways

  • When the elevator moves at constant velocity or is stationary, the apparent weight equals the true weight.
  • When the elevator accelerates upward, apparent weight increases.
  • When the elevator accelerates downward, apparent weight decreases.
  • Newton’s second law is key in calculating the apparent weight in accelerating systems.

Join the Discussion

Elevator problems provide a great way to understand how forces work in non-inertial frames! Have any questions or tips on solving these problems? Drop a comment below, and let’s discuss!

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