When a mass of 25 kg is hung from the middle of a fixed...

Key Concepts

Static Equilibrium

This concept involves ensuring that all forces and moments acting on a system cancel out so that the system remains at rest. In the context of a suspended wire or cable, static equilibrium is used to set up equations where the vertical forces (such as the weight of a load) are balanced by the vertical components of the tension in the wire, while horizontal components cancel out because of symmetry. This principle is foundational for analyzing any system at rest.

Force Decomposition

Force decomposition is the process of breaking a force into perpendicular components, typically horizontal and vertical. In problems involving a sagging wire, the tension force that acts along the wire is decomposed into components using the given sag angle. This decomposition allows for the establishment of equilibrium equations by matching the vertical component with half of the load on each side and ensuring the horizontal components are balanced.

Tension in Cables

Tension refers to the pulling force transmitted along a cable, string, or wire when it is subjected to a load. In a sagging cable problem, understanding how the tension varies and its relation to the angle of sag is crucial. The tension is the key force that, when resolved into its components, must counteract the gravitational force exerted by the hanging mass.

Geometry of Circular Arcs and Radius of Curvature

When a wire or cable sags under an applied load, it often approximates the shape of a circular arc, at least locally. The radius of curvature of that arc quantifies how sharply the cable bends; a larger radius means a gentler curve while a smaller radius implies a tighter bend. Establishing the relationship between the sag angle, the tension force, and the curvature allows for the calculation of this radius, linking mechanical properties with geometric analysis.

Trigonometric Relationships

Trigonometric functions such as sine and cosine are essential in relating the angles and force components in static problems. When a wire sags to form an angle with the horizontal, these relationships help in decomposing the tension force into its horizontal and vertical parts. This step is crucial for setting up and solving the equilibrium equations necessary to determine quantities like the radius of curvature.

Transcript

00:01 All right, problem 965.

00:03 Chapter 9, problem 65.

00:06 And so here's the question.

00:08 When a massive 25 kilograms is hung from the middle of a fixed straight aluminum wire, the wire sags to make an angle of 12 degrees with the horizontal as shown in figure 9 -83.

00:19 It determined the radius of the wire.

00:22 Okay, so our goal is to find the radius of the wire, and since it has a radius, we can assume that there's a circle involved.

00:29 If there's a circle involved with a long material like a rope or a wire, it's probably going to be cylindrical, so you can just assume that for the wire.

00:44 To find r, the radius of the wire, we're going to use the relationship between stress and strain, which you should have seen in section 9 of the book in part 5, or section 5, sorry.

00:55 So this equation tells us that the force divided by the area is equal to young's modulus multiplied by the ratio of the length.

01:05 Well, the ratio of the change in length versus the original length.

01:12 So one thing that is not immediately obvious, but maybe you should know that the area of a circle or the face of a cylinder is going to be pi r squared.

01:27 So we can replace a with pi r squared.

01:32 And then all we have to do is, well, now that radius is involved in the equation, we can just, now that radius is involved in the equation, since it wasn't before, we can rearrange the equation and have r on one side by itself, a single r, and then all the rest of the stuff on the other side, and that should help us find the radius.

01:54 So what is given to us? e is given to us e being young's modulus as 70 times 10 to the 9 over or newton's per meter squared and so just to talk about a little bit about what that is so young's modulus is the elastic moduli and essentially it's just a constant of proportionality so it helps you it kind of gives you a gauge of what different materials you're working with in a problem because some things may be more vulnerable to stress and strain than other things.

02:35 Some materials may be more vulnerable to stress and strain than others.

02:39 So you may be able to rip, you know, a tissue paper very easily, but maybe not so much a phone book, something like that.

02:48 So this is why we need to use young's modulus to scale the different materials to the correct forces.

02:57 But anyway, we're given that, and we also know what pie is.

03:01 Can count that given.

03:03 But what's not given to us that we need to find the radius.

03:06 Of course, radius is included there, but i didn't write it down.

03:09 We need the ratio of the lengths and we need the force.

03:12 So how do we find those two things? well, we're going to figure that out.

03:17 Next slide.

03:18 So for force, we just want to use a free body diagram.

03:22 So what you can do is choose the attachment point as your junction.

03:25 So where all the wires are meeting up together and holding that weight as your junction and then just look below.

03:31 And so if you focus on these three arrows pointing out from the center just real quick, just to understand that.

03:38 So we have the tension force in one direction to the left and the tension force on the right.

03:43 So two tensions there.

03:45 And then we have mg downwards just forced due to gravity, otherwise known as the weight.

03:51 But what you can do after you have this free body diagram is you can start summing the forces.

03:58 And so when you sum the forces and use newton's second law, this can help us find a force, basically.

04:03 So we can start with x or y.

04:06 Let's start with x...

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