Problem 52 - College Physics

Question

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) \(725 \mathrm{N},(\mathrm{b}) 595 \mathrm{N}\)

Step-by-Step Solution

Answer

(a) 0.60 m/s² upwards, (b) 1.27 m/s² downwards.

Step 1: 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.

Step 2: 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.

Step 3: 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.

Step 4: Calculate Mass

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

Step 5: 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.

Step 6: 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.

Key Concepts

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/s² 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/s² ≈ 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 × Gravity

Here, gravity is approximately 9.81 m/s². Regardless of the elevator's motion, gravity’s 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.