You're riding a Ferris wheel while sitting on a scale. (a) Why does the scale reading when you'r

You're riding a Ferris wheel while sitting on a scale. (a)...

Key Concepts

Normal Force

This is the force exerted by a surface that prevents an object from accelerating through it. In the context of a scale reading, the normal force is what the scale measures, and it can vary when additional accelerations, such as those from circular motion, are present.

Centripetal Acceleration

In circular motion, centripetal acceleration is the inward acceleration that keeps an object moving along a curved path. It affects the net force acting on an object, causing variations in the apparent weight measured by the scale when the object is at different points along the path.

Free-Body Diagrams

These are graphical representations of the forces acting on an object. They are used to visualize and analyze how forces such as gravity, normal force, and any additional forces due to acceleration interact as the object moves along its path.

Newton's Second Law in Circular Motion

Newton's Second Law (F = ma) is applied in circular motion to account for the net force toward the center of the circle (centripetal force). This law helps explain how the apparent weight changes depending on whether the centripetal force adds to or subtracts from the gravitational force.

Transcript

00:02 When you're on a ferris wheel, it goes around at a constant rate once it's sort of up to speed.

00:08 So it doesn't matter whether you're at the top or the bottom, you're going at a fixed constant speed.

00:16 And according to what we're given in this problem, this ferris wheel has a radius of 9 .6 meters and a time of rotation of 36 seconds, 9 .6 meters.

00:38 And we're asked to a couple things.

00:40 One, why is the scale reading a different value when you're at the top and the bottom? so let's say you're in the thing and you're sitting on a scale, and you're sort of measuring your quote -unquote weight.

00:49 It seems to vary between the top and the bottom.

00:53 And then we'll follow through on that by doing some calculations to find and what is the scale reading at the top and the bottom.

01:02 So at the top, well, anywhere on this thing, you're moving in a circle.

01:07 And so this being moving in a circle at a constant rate, we're talking about circular motion and centripetal force.

01:15 And the centripetal force is mv squared over r.

01:21 The centripetal acceleration, that all equals ma.

01:26 So we'll call that ac.

01:27 So that means that the centripetal acceleration equals v squared over r, which we're going to use pretty shortly.

01:35 But what may be more important at the moment is the fact that when you're moving in a circle, the centripetal force always points towards the center.

01:46 And the centripetal acceleration always exists towards the center.

01:51 If you're at the bottom, you're going to accelerate upward.

01:53 If you're at the top, you're accelerating downward.

01:55 You're always accelerating towards the center of the circle.

01:58 And so that's going to dictate how we address the forces at the top and the bottom.

02:03 So let's look at the forces at the top.

02:07 At the top, we've got m .g down and a normal force up since you're sitting on the seat.

02:14 You're not flying out of the seat.

02:15 So a normal force does exist.

02:18 But because the acceleration is downward, we're going to call downward the positive direction for our analysis at that point.

02:28 So in any case with newton's second law, we've got the sum of the forces equals ma.

02:37 Well, in this case, the sum of the forces pointing downward, m g is the positive force.

02:45 So we've got m g minus the normal force equals m a.

02:51 Now at the bottom, the same forces exist.

02:59 There's m g down and there's a normal force up.

03:03 But when you're at the bottom, your acceleration is upward.

03:07 So that's going to change our signage in how we write the sum of the forces.

03:12 At this moment, the normal force is the positive force.

03:16 M .g is the negative force, and those sum up to equal m .a.

03:22 So when we're asked, why do you feel heavier? why does the scale read something different? really, what that's asking about is the normal force, the amount of squish between your bottom and the seat that you're sitting on.

03:39 And so let's rearrange each of these to solve for the normal force.

03:43 So if we move mg over the other side, so we get negative n equals m -a minus m -g.

03:53 If we rearrange that, multiply through by negative 1, we get the normal force equals m -g minus m -a...

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