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Chapter 4: Problem 45

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.

Short Answer

Expert verified
The elevator accelerates upwards at 0.60 m/s虏 with a 725 N reading and downwards at 1.26 m/s虏 with a 595 N reading.

Step by step solution

01

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.
02

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 \( m = \frac{683}{9.8} \approx 69.7 \text{ kg} \).
03

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 \( \text{ m/s}^2 \).
04

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 \( \text{ m/s}^2 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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/s虏.

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/s虏 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鈥檇 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.

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Most popular questions from this chapter

An 8.00-kg box sits on a level floor. You give the box a sharp push and find that it travels 8.22 m in 2.8 s before coming to rest again. (a) You measure that with a different push the box traveled 4.20 m in 2.0 s. Do you think the box has a constant acceleration as it slows down? Explain your reasoning. (b) You add books to the box to increase its mass. Repeating the experiment, you give the box a push and measure how long it takes the box to come to rest and how far the box travels. The results, including the initial experiment with no added mass, are given in the table: In each case, did your push give the box the same initial speed? What is the ratio between the greatest initial speed and the smallest initial speed for these four cases? (c) Is the average horizontal force \(f\) exerted on the box by the floor the same in each case? Graph the magnitude of force \(f\) versus the total mass \(m\) of the box plus its contents, and use your graph to determine an equation for \(f\) as a function of \(m\).

Crates \(A\) and \(B\) sit at rest side by side on a frictionless horizontal surface. They have masses \(m_A\) and \(m_B\), respectively. When a horizontal force \(\vec{F}\) is applied to crate \(A\), the two crates move off to the right. (a) Draw clearly labeled free-body diagrams for crate \(A\) and for crate \(B\). Indicate which pairs of forces, if any, are third-law action-reaction pairs. (b) If the magnitude of \(\vec{F}\)is less than the total weight of the two crates, will it cause the crates to move? Explain.

The froghopper (\(Philaenus spumarius\)), the champion leaper of the insect world, has a mass of 12.3 mg and leaves the ground (in the most energetic jumps) at 4.0 m/s from a vertical start. The jump itself lasts a mere 1.0 ms before the insect is clear of the ground. Assuming constant acceleration, (a) draw a free-body diagram of this mighty leaper during the jump; (b) find the force that the ground exerts on the froghopper during the jump; and (c) express the force in part (b) in terms of the froghopper's weight.

A chair of mass 12.0 kg is sitting on the horizontal floor; the floor is not frictionless. You push on the chair with a force \(F =\) 40.0 N that is directed at an angle of 37.0\(^\circ\) below the horizontal, and the chair slides along the floor. (a) Draw a clearly labeled free-body diagram for the chair. (b) Use your diagram and Newton's laws to calculate the normal force that the floor exerts on the chair.

A small 8.00-kg rocket burns fuel that exerts a timevarying upward force on the rocket (assume constant mass) as the rocket moves upward from the launch pad. This force obeys the equation \(F = A + Bt^2\). Measurements show that at \(t\) = 0, the force is 100.0 N, and at the end of the first 2.00 s, it is 150.0 N. (a) Find the constants \(A\) and \(B\), including their SI units. (b) Find the \(net\) force on this rocket and its acceleration (i) the instant after the fuel ignites and (ii) 3.00 s after the fuel ignites. (c) Suppose that you were using this rocket in outer space, far from all gravity. What would its acceleration be 3.00 s after fuel ignition?
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