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 recall that your normal weight is $625 \mathrm{~N}$. Draw a free-body diagram. (a) When the elevator has an upward acceleration of magnitude $2.50 \mathrm{~m} / \mathrm{s}^{2},$ what does the scale read? (b) If you hold a $3.85 \mathrm{~kg}$ package by a light vertical string, what will be the tension in this string when the elevator accelerates as in part (a)?

Key Concepts

Free-Body Diagram

A diagram that illustrates an object isolated from its surroundings with all the forces acting upon it. It is used to visually sum up forces like gravity, normal force, tension, and others, in order to apply Newton’s laws of motion correctly.

Newton’s Second Law

A fundamental principle stating that the net force acting on an object is equal to the mass of the object multiplied by its acceleration. This concept is key when analyzing the motion of objects in accelerating systems such as elevators.

Normal Force as Apparent Weight

In accelerated systems like an elevator, the scale reading (apparent weight) is the normal force, which may differ from the gravitational weight if the elevator is accelerating. This adjusted reading results from the additional force components due to acceleration.

Tension in a String

The force that is transmitted through a string when it is used to support or move an object. In an accelerating frame, calculating tension involves accounting for both the gravitational force and the additional force due to the frame’s acceleration.

Accelerated Frames and Pseudo-Force

In non-inertial (accelerating) frames, an extra fictitious force, known as a pseudo-force, must be introduced to account for the apparent acceleration effects. This concept is essential for correctly applying Newton’s laws in accelerating systems like elevators.

Transcript

00:01 Alright, so this is a fun little problem involving scales inside of elevators.

00:07 So the most important piece of information we need is your weight, your hypothetical weight.

00:14 And this problem, which we're saying, is 625 newtons.

00:19 So they don't tell us what your mass is, which will also be useful later.

00:22 So let's just go ahead and solve for that right now.

00:25 So you can find your mass by defining your weight, by the gravitational accelerators.

00:30 Near the surface of the earth, which is 9 .81 meters per square second.

00:38 And then we find that our mass, which we use the problem, is 63 .7 kilograms.

00:51 Good, good, good.

00:52 So, we can now begin the problem, and even before we begin, it tells us to draw a free body diagram, which is a good exercise to do.

01:03 So this is a picture of you.

01:04 You are a box and inside an elevator there are two forces that are acting upon.

01:09 One is of course your weights from the omnipresent force of gravity and the other is a normal vector, a normal meaning perpendicular and the basically the floor pushing back up against you so you don't go falling for the elevator.

01:25 So that's normally an equilibrium except when the accelerator accelerates when it starts to move.

01:32 So in our problem, we're saying that the elevator is accelerating at a rate of 2 .50.

01:41 So that's an a for acceleration 2 .50 meters per square second in the upward direction.

01:52 So in other words, the net force acting on you is going to be, and i'm getting this from the three body diagram.

02:00 The normal force minus your weight.

02:02 And that is also going to be equal to m .a as we can the second log tells us.

02:07 So what is the normal force needed to achieve the acceleration of 2 .50 meters per second? well, let's solve for it.

02:16 Let's add w to both sides of this equation, so that we end up with n is equal to weight plus m .a...

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