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Chapter 4: Problem 52

You leave the doctor's office after your annual checkup and recall that you weighed 683 \(\mathrm{N}\) in her office. 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) \(725 \mathrm{N},(\mathrm{b}) 595 \mathrm{N}\)

Short Answer

Expert verified
(a) 0.60 m/s虏 upwards, (b) 1.27 m/s虏 downwards.

Step by step solution

01

Understanding the Problem

We are given a person's weight in a doctor's office, measured as 683 N, which is the gravitational force acting on the person (their weight). In an elevator, the scale measures different weights due to the elevator's acceleration. We need to find the magnitude and direction of acceleration of the elevator based on these different scale readings.
02

Identify Known Quantities

Your weight in the doctor's office is given as the normal weight when the elevator has zero acceleration, which is 683 N. We have two different readings from the scale in the elevator: 725 N when accelerating in one direction, and 595 N when accelerating in another direction.
03

Use Newton's Second Law

Newton's second law states that the net force acting on an object is equal to the mass of the object multiplied by its acceleration: \[ F = ma \]Here, the force exerted by the scale equals your apparent weight. The actual force of gravity on you (your normal weight) is 683 N.
04

Calculate Mass

First, we calculate your mass using your weight when not accelerating (683 N):\[ m = \frac{F}{g} = \frac{683 \text{ N}}{9.81 \text{ m/s}^2} \approx 69.6 \text{ kg} \]
05

Find Acceleration when Scale Reads 725 N

For the case when the scale reads 725 N:1. The apparent force (F') = 725 N2. Use the equation F' = m(g + a) to solve for a:\[ 725 = 69.6 \, (9.81 + a) \]3. Solve for a:\[ a = \frac{725}{69.6} - 9.81 \approx 0.60 \, \text{m/s}^2 \]Since the scale reads more than your weight, the acceleration is upwards.
06

Find Acceleration when Scale Reads 595 N

For the case when the scale reads 595 N:1. The apparent force (F") = 595 N2. Use the equation F" = m(g - a) to solve for a:\[ 595 = 69.6 \, (9.81 - a) \]3. Solve for a:\[ a = 9.81 - \frac{595}{69.6} \approx 1.27 \, \text{m/s}^2 \]Since the scale reads less than your weight, the acceleration is downwards.

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Key Concepts

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

Weight Measurement
Understanding weight measurement is crucial when analyzing forces acting on an object. Weight is simply the force of gravity acting on an object's mass. This force is calculated by multiplying the object's mass by the acceleration due to gravity, which is approximately 9.81 m/s虏 on Earth.
For example, if you weigh 683 N in a doctor's office, this means the gravitational force acting on your mass is 683 N. Your mass can be found by dividing your weight by the acceleration due to gravity:
  • Mass = Weight / Gravity
  • Mass = 683 N / 9.81 m/s虏 鈮� 69.6 kg
Weight measurements can change in scenarios where acceleration is involved, such as in an elevator. Here, the apparent weight is different based on the direction and magnitude of acceleration.
Elevator Acceleration
When an elevator moves, it causes a change in acceleration that affects how heavy or light we feel. This is due to Newton's Second Law, which tells us the net force on an object is the product of its mass and acceleration:
  • F = ma
In an elevator, if the acceleration aligns with gravity, your apparent weight changes. For instance:
  • If you're accelerating upwards, you feel heavier because the net force increases.
  • If you're accelerating downwards, you feel lighter as the net force decreases.
These shifts are why an elevator's acceleration can make a weight scale show a different reading than when you're stationary.
Gravitational Force
Gravitational force is a natural phenomenon by which all things with mass or energy are brought toward one another. On Earth, this force gives weight to physical objects. It is computed as the product of an object's mass and the Earth's gravitational pull:
  • Gravitational Force = Mass 脳 Gravity
Here, gravity is approximately 9.81 m/s虏. Regardless of the elevator's motion, gravity鈥檚 role as the constant background force stays the same. This means when measuring weight on a scale stationary vs. moving upward or downward, the fundamental gravitational pull on the body remains constant.
Apparent Weight
Your apparent weight is what you perceive as your weight when experiencing additional forces, such as acceleration in an elevator. It reflects how heavy or light you feel due to these additional forces modifying the gravitational pull.
In the elevator example:
  • When the scale reads 725 N, the elevator accelerates upwards, creating a force in addition to gravity, increasing apparent weight.
  • When the scale reads 595 N, the elevator accelerates downwards, opposing gravity, reducing apparent weight.
These variances in apparent weight occur because the scale measures the normal force it exerts on you, which changes with elevator acceleration. This is different from actual weight, which only involves gravitational force.

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Most popular questions from this chapter

(a) What is the mass of a book that weighs 3.20 \(\mathrm{N}\) in the laboratory? (b) In the same lab, what is the weight of a dog whose mass is 14.0 \(\mathrm{kg} ?\)

Two dogs pull horizontally on ropes attached to a post; the angle between the ropes is \(60.0^{\circ} .\) If \(\operatorname{dog} A\) exerts a force of 270 \(\mathrm{N}\) and dog \(B\) exerts a force of \(300 \mathrm{N},\) find the magnitude of the resultant force and the angle it makes with dog \(A\) 's rope.

Human biomechanics. The fastest pitched baseball was measured at 46 \(\mathrm{m} / \mathrm{s}\) . Typically, a baseball has a mass of 145 \(\mathrm{g}\) . If the pitcher exerted his force (assumed to be horizontal and constant) over a distance of \(1.0 \mathrm{m},(\mathrm{a})\) what force did he produce on the ball during this record-setting pitch? (b) Make free-body diagrams of the ball during the pitch and just after it has left the pitcher's hand.

Interpreting a medical chart. You, a resident physician, are reading the medical chart of a normal adult female patient. Carelessly, one of the nurses has entered this woman's weight as a number without units. Another nurse has offered a suggestion for what the units might be. In each of the following cases, decide whether this nurse's suggestion is physically reasonable: (a) The number is \(150,\) and the nurse suggests that the units are kilograms. (b) The number is \(4.25,\) and the nurse suggests that the units are slugs. (c) The number is \(65,000,\) and the nurse suggests that the units are grams.

A warehouse worker pushes a crate along the floor, as shown in Figure \(4.33,\) by a force of 10 \(\mathrm{N}\) that points downward at an angle of \(45^{\circ}\) below the horizontal. Find the horizontal and vertical components of the push.
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