Question

Evaluate If you put a scale in an elevator and weigh yourself as you ascend and then descend, does the scale have the same reading in both instances? Explain your answer.

Short answer

The scale will not have the same readings while ascending and descending in an elevator. This is because the apparent weight acting on a person depends on the acceleration of the elevator. While ascending, the elevator accelerates upward, increasing the apparent weight, resulting in a higher reading on the scale. In contrast, while descending, the elevator accelerates downward, decreasing the apparent weight, resulting in a lower reading on the scale. Therefore, the readings on the scale differ in both instances.

Step-by-step solution

Understand Apparent Weight and Forces on Body

Apparent weight is the force a body experiences due to gravity and acceleration. It is the force that a body exerts on a supporting surface (in this case, the scale). Apparent weight can be determined by the equation: \( W_{apparent} = mg + ma \) where: - \(W_{apparent}\) = apparent weight - \(m\) = mass of the person - \(g\) = acceleration due to gravity (approximately \(9.81 m/s^2\)) - \(a\) = acceleration of the elevator (upward is positive, downward is negative) Now, let's analyze each scenario (ascending and descending) separately.

Scenario 1: Ascending

While the elevator is ascending, it is accelerating in the upward direction. This means, \(a\) is positive. From the apparent weight formula, we get: \( W_{ascending} = mg + ma \) Since \(a > 0\), the total apparent weight is increased from the normal weight (when the elevator is stationary). Thus, when ascending, the scale will show a higher reading.

Scenario 2: Descending

While the elevator is descending, it is accelerating in the downward direction. This means, \(a\) is negative. From the apparent weight formula, we get: \( W_{descending} = mg + ma \) Since \(a < 0\), the total apparent weight is decreased from the normal weight (when the elevator is stationary). Thus, when descending, the scale will show a lower reading.

Compare the Apparent Weights

Now that we have determined the apparent weight while ascending and descending, let's compare the readings: \[ W_{ascending} = mg + ma_{up} \] \[ W_{descending} = mg + ma_{down} \] We can clearly see that the two readings are different because the acceleration values used, \(a_{up}\) and \(a_{down}\), are different in both cases. Thus, the answer to the question is no, the scale will not have the same readings while ascending and descending in an elevator.

Key concepts

Elevator Physics

Elevator physics is fascinating, especially when you think about the forces at play when you're riding in one. Imagine stepping into an elevator with a scale. As you ascend or descend, the reading on the scale changes. This change is due to the apparent weight, which varies with the elevator's motion.

When the elevator moves, it doesn't just affect your position; it influences the net forces acting on you, altering what we call your apparent weight. This makes elevator physics a great practical example to understand how forces work beyond simple stationary scenarios.

In short:

• Ascending increases apparent weight.
• Descending decreases apparent weight.

Analyzing these conditions provides a deeper understanding of how forces like gravity and acceleration interplay in real-life situations.

Acceleration

Acceleration is a key concept when discussing elevator physics. It refers to the change of velocity of an object over time. In the case of an elevator, acceleration is what causes your apparent weight to change.

When the elevator speeds up or slows down, it accelerates in the respective direction. If it's ascending, it accelerates upwards, resulting in a higher reading on the scale due to added force. Conversely, when descending, the elevator accelerates downward, which results in a lower scale reading.

Important points to remember:

• Upward acceleration → apparent weight increases.
• Downward acceleration → apparent weight decreases.

This alteration in apparent weight during acceleration is vital for understanding everyday experiences and deeper physics concepts such as force and motion.

Newton's Laws

Newton's laws of motion provide the foundation for understanding the forces involved in elevator physics. Let's focus on the second law which is particularly significant in this context.

The second law states: \[ F = ma \]where:

• \( F \) is the net force,
• \( m \) is the mass,
• \( a \) is the acceleration.

For a person in an elevator, this law explains how acceleration affects the forces we experience. The force you feel, or your apparent weight, changes with the elevator's motion.

• If acceleration is upwards, the net force is greater, increasing your apparent weight.
• If acceleration is downwards, the net force is less, reducing your apparent weight.

Exploring this within the context of elevator motion illustrates Newton’s laws in action, helping solidify understanding of fundamental physics principles.

Gravity

Gravity is the invisible force drawing us towards the Earth, and it plays a crucial role in elevator physics by constantly exerting a downward force on you. This gravitational pull contributes to your standard weight when you stand still.

In the equation for apparent weight, \( mg \) represents the gravitational force acting on you, where \( m \) is your mass, and \( g \) is the acceleration due to gravity (around \( 9.81 \, m/s^2 \)).

• Gravity remains constant, irrespective of the elevator moving or not.
• It provides the baseline weight which is then modified by any elevator-induced accelerations.

The interaction between constant gravity and the variable acceleration of the elevator is what results in the different scale readings during ascent and descent. Understanding gravity's role helps appreciate how these forces interact during everyday scenarios like using an elevator.