Problem 45 - University Physics with Modern Physics

Question

After an annual checkup, you leave your physician's office, where you weighed 683 N. You then get into an elevator that, conveniently, has a scale. Find the magnitude and direction of the elevator's acceleration if the scale reads (a) 725 N and (b) 595 N.

Step-by-Step Solution

Answer

The elevator accelerates upwards at 0.60 m/s² with a 725 N reading and downwards at 1.26 m/s² with a 595 N reading.

Step 1: Understand the Problem

We need to find the acceleration of an elevator based on the difference in weight readings when standing on a scale inside it. The change in weight indicates an acceleration different from gravity. The person's weight on the ground is 683 N. In the elevator, the scale reads differently, indicating either additional or less force due to the elevator's motion.

Step 2: Determine the Gravitational Force

The gravitational force acting on the person is their weight on the ground, 683 N. Since weight is calculated by the formula \( F = mg \), where \( m \) is mass and \( g \) is acceleration due to gravity, we can find the mass \( m \) by rearranging to \( m = \frac{F}{g} \). Given \( g = 9.8 \text{ m/s}^2 \), we find \( m = \frac{683}{9.8} \approx 69.7 \text{ kg} \).

Step 3: Analyze Scale Reading of 725 N

When the elevator's scale reads 725 N, the net force on the person is the reading (725 N) minus the gravitational force, 683 N. The net force, \( F_{net} \), is 42 N upwards (since 725 N > 683 N, indicating an upward acceleration). Using Newton's second law \( F = ma \), we find \( a = \frac{F_{net}}{m} = \frac{42}{69.7} \approx 0.60 \text{ m/s}^2 \). Thus, the elevator accelerates upwards at 0.60 \( \text{ m/s}^2 \).

Step 4: Analyze Scale Reading of 595 N

In this case, when the scale reads 595 N, the net force is \( 595 - 683 = -88 \text{ N} \). The negative sign indicates a downward acceleration. Using \( F = ma \), the acceleration \( a = \frac{-88}{69.7} \approx -1.26 \text{ m/s}^2 \). Thus, the elevator accelerates downwards at 1.26 \( \text{ m/s}^2 \).

Key Concepts

Elevator Physics

When we step into an elevator, the experience often feels a bit like magic, but it's actually a simple physics concept in action. As the elevator moves, it accelerates either upwards or downwards. You might notice this when your stomach flutters or you feel a bit heavier or lighter than usual. This sensation happens because of changes in forces acting on your body as the elevator moves.

Inside an elevator, a scale will show a different weight due to these changes. This happens because the scale measures the support force needed to keep you stationary with respect to the elevator's floor. As the elevator accelerates, this support force changes, altering the scale reading.

• If the elevator accelerates upwards, the scale reads a greater weight. This is because the elevator adds to the gravitational force, requiring more support force to hold you up.
• If the elevator accelerates downwards, the scale reads a lesser weight. This is because the gravitational pull is only partially counteracted, leading to a reduction in the support force needed.

Understanding how forces interact in an elevator is not only fascinating but helps us appreciate the real-world application of Newton's laws.

Gravitational Force

Gravitational force is one of the most fundamental forces in nature. It's the force that pulls objects towards the center of the Earth, keeping us grounded. This force is directly related to an object's mass and the acceleration due to gravity, which on Earth is approximately 9.8 m/s².

The formula used to calculate gravitational force is simple:\[ F = mg \]where:

• \( F \) is the force in newtons (N)
• \( m \) is the mass in kilograms (kg)
• \( g \) is the acceleration due to gravity (9.8 m/s² on Earth)

Thus, a person's weight is just the gravitational force acting on their mass. When in an elevator, if only gravitation acted, you’d always weigh the same. But since the elevator accelerates, a different force acts upon you, modifying the weight the scale measures. This results in longer or shorter arrow readings, indicating greater or lesser force than just gravity alone.

Net Force

In physics, the net force is the sum of all forces acting on an object. When it comes to elevator scenarios, net force is crucial in determining motion. The concept can be explained easily through Newton's Second Law of Motion, which states that force equals mass times acceleration (\[ F = ma \]).

When you are standing on a scale in an elevator that is accelerating, the forces at play include both the gravitational force and the additional force due to the elevator's movement.

• If the net force is positive, it indicates upward acceleration, making you feel heavier.
• If the net force is negative, it implies downward acceleration, giving the sensation of lightness.

The net force explains why a person's weight differs when stepping into an elevator in motion. By understanding the magnitude and direction of this net force, you can determine the elevator's acceleration whether it's going up or down. This calculation helps illustrate how we can experience altered forces in daily life, applying fundamental physics principles.