You walk into an elevator, step onto a scale, and push the...

You walk into an elevator, step onto a scale, and push the "up" button. You also recall that your normal weight is 625 $\mathrm{N}$ . Start each of the following parts with a free-body diagram. (a) If the elevator has an acceleration of magnitude $2.50 \mathrm{m} / \mathrm{s}^{2},$ what does the scale read? (b) If you start holding a 3.85 kg package by a light vertical string, what will be the tension in this string once the elevator begins accelerating?

Key Concepts

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. This principle is crucial for analyzing situations where additional acceleration modifies the forces acting on a body, resulting in changes to measures such as apparent weight or tension in a supporting string.

Free-Body Diagrams

A free-body diagram is a graphical illustration used to visualize the forces acting on a single object. It simplifies the process of applying Newton's laws by isolating the object and representing all external forces as vectors, which is essential in analyzing systems undergoing acceleration, such as an elevator.

Apparent Weight and Normal Force

Apparent weight refers to the normal force experienced by an object, which can differ from the object's true weight when the system is accelerating. In non-inertial frames, like an accelerating elevator, the apparent weight is calculated by considering both gravitational force and the additional force due to acceleration, leading to an increased or decreased reading depending on the direction of acceleration.

Tension in Accelerated Systems

Tension in a string supporting an object is affected by the acceleration of the system. When the system accelerates, the tension must not only balance the gravitational force but also provide the extra force required for the object's acceleration. This concept is directly derived from Newton's Second Law and is integral to solving problems involving objects attached by strings in non-inertial frames.

Transcript

00:03 Alright, so in this problem, we have an elevator and a man or a whole person in this elevator standing on some type of scale and he hits the button that makes the elevator go up and he's accelerating up a 2 .5 meters per second squared and his his normal weight is 625 newtons.

00:46 Do do do do do do okay so that's our problem we want to know with it accelerating what this scale reads well first what we need to know is we need to understand the mass of this person um because i want to calculate the force of this vertical acceleration and so i'm going to do 625 newtons and divide that by 9 .8 meters per second squared that will get me um that will give me the weight and the mass in kilograms.

01:26 And then i'm gonna take that mass and i'm gonna multiply it times 2 .5 meters per second squared to convert it back into new newton measurement.

01:38 And i'm gonna cheat and use my scratch work because i already did this in a calculator earlier.

01:42 And i got a, uh, do to do to do.

01:46 Ignore my notes here a second, i'm sorry.

01:50 A 159 .4, right.

01:52 Okay.

01:52 So if you want to 81, 59 .4.

02:02 So if you want to know the weight on that scale, well, let's draw this drawing a little bigger.

02:09 Here's my scale.

02:15 In the down direction, i have the person's weight, which is 625 newton's.

02:24 But in the upward direction, how to do it say this.

02:42 So the scale's moving upward.

02:51 At 159 .4.

02:57 We don't know the mass of the scale.

02:59 How to say this? let me draw this better.

03:25 Okay, here we go.

03:27 This person is being, the person on the scale is being accelerated at 159 .4.

03:38 So not only, but, so that means that not only is their weight, not only is the person's weight being sent back onto the scale, but we also have, if the scale, if this person is being pushed upward at this force, this is my force, then it's going to send an equal and opposite force back into the scale.

03:59 So there's also, this is going to send a equivalent equal opposite back into the scale.

04:11 That means that our, do to do, means that our scale has two, has a downward force being put on it.

04:28 The downward force being put on our scale is equal to the weight, plus the normal force of the person's, yeah, the normal force of the scale.

05:18 Because this whole elevator is pushing this person up at 159 .4 newtons.

05:22 That means that there's normal response backwards.

05:24 What is that response? push against the scale.

05:27 So we add the two...