(II) Find the tension in the two cords shown in Fig. 9-52. Neglect the mass of the cords, and as

step 1 Once again, welcome to a new problem. This time, we want to find attention. We're given a struct

(II) Find the tension in the two cords shown in Fig. 9-52....

Key Concepts

Trigonometric Relationships in Mechanics

Trigonometric functions, such as sine and cosine, are vital in converting angled forces into their perpendicular components. In the context of tension problems, these relationships allow us to relate the applied forces to the angles at which they act. This is essential for accurately setting up the equations that govern the balance of forces in static equilibrium.

Vector Resolution of Forces

When forces act at angles, it is essential to break them down into their horizontal and vertical components using vector resolution. This process involves the use of sine and cosine functions to determine the magnitudes of the components, which are then used in equilibrium equations. This concept is critical in problems where forces are applied at an angle relative to the coordinate axes.

Free-Body Diagram

A free-body diagram (FBD) is a graphical representation that depicts all external forces acting on an isolated body. Drawing an FBD helps in visualizing the problem, identifying the directions and magnitudes of the forces, and setting up the equations of equilibrium. It is a fundamental tool in mechanics for solving problems involving multiple forces, including those that act at various angles.

Static Equilibrium

This concept involves analyzing a system where the sum of all forces and torques acting on an object is zero. In problems involving suspended masses, static equilibrium is used to set up equations that ensure the object is at rest, meaning that upward tensions balance the downward gravitational force. This principle is fundamental in calculating unknown forces in the system.

Tension in Cords

Tension is the pulling force transmitted along a rope, string, or cord. When a mass is suspended by cords at an angle, the tension in each cord must be considered as part of the overall balance of forces acting on the mass. Understanding tension and its distribution in multiple cords is crucial to solving for the unknown forces, particularly when the cords are assumed to be massless.

Transcript

00:01 Once again, welcome to a new problem.

00:05 This time, we want to find attention.

00:08 We're given a structure almost like a kind of lever, but not exactly.

00:17 I mean, it's a plunk that looks like that, horizontal plank supported by a vertical plank or like a small column.

00:32 And it forms the kind of liver there's a string hanging off of the kind of the kidney lever right here and it's holding a weight that's hanging from this location okay there's a weight m you know this weight is m and also we're given an angle theta theta happens to be 33 degrees and m and is equivalent to 190 kilograms right away we can find the weight of this object remember acceleration produces weight here we're dealing with acceleration due to gravity which is g so we're going to take 190 kilograms times 9 .8 meters per second squared or we could say 9 .81 meters per second squared if we punch in those numbers we get you end up and up getting hundred and ninety times nine point eight one that's going to give us let's see one ninety times nine point eight one thousand one thousand eight hundred and sixty three point nine newtons remember weight is a force so we have to compute that in terms of newton's that's the information we're given the goal of this problem is to figure out what these two forces are so you know there's a force there's a tension there's a uniform tension in the string so you know this one is pulling that way and that one is pulling that way this one we're calling it f1 and that one's f2.

02:43 So that's our goal.

02:46 Remember, we're told that the codes, they don't have a mass, so we can say they're massless, you know, massless codes.

02:59 So that's our goal.

03:00 You know, we want to find out the tension.

03:04 You know, we want to find out the tension in the string.

03:09 We could actually, instead of calling it f4, and f2 we could call it t1 and t2 just to represent tension so this is uh t1 and t2 you know they're pulling that way that's the information given you know that's the information we're given so we'll have like a free body diagram that represents these relationships uh at the center is like uh connector we have a mass pointing downwards this is m g that's the weight of this block right here and then there's a tension pulling that way to the right we'll call that t2 and then there's a you know we can make right the positive and then there's another tension there's a tension pulling that way there's a tension pulling that way pulling at an angle right there and you know some of the things we can do is if you if this angle is theta then this angle will also be theta that angle that small angle so this one will also be data so there's a we can resolve this into two components this is the tension the first tension t1 and we resolve it two components, the vertical upwards components, this is positive, is t1, sign of theta, and then the component to the right, you can think of that as on the negative side is t1 cosine of theta.

05:05 All we did is just to resolve those components.

05:09 So we'll deal with the tension along the x -axis, so we'll say the sum of the tensions.

05:16 Or some of the forces in the x is zero so that means t2 minus t1 cosine of theta is going to be zero and that helps us have an algebraic solution for t2 that's dependent on t1 and the angle it makes with the horizontal which is theta and so we also also, on the next page, we'll see that if you look at this horizontal, this t1 sign of theta is holding the weight mg because there's equilibrium.

06:03 So we'll say the sum of tensions in the wide direction is zero.

06:10 T1 sign of theta is pointing upwards, so minus mg is zero...

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