The three ropes in Figures Ex 6.1 are tied to a small, very...

The three ropes in Figures Ex 6.1 are tied to a small, very light ring. Two of the ropes are anchored to walls at right angles, and the third rope pulls as shown. What are $T_{1}$ and $T_{2},$ the magnitudes of the tension forces in the first two ropes? (FIGURES CAN'T COPY)

Key Concepts

Tension Forces

Tension forces are the forces transmitted through a rope, cable, or similar connector when it is pulled tight by forces acting from opposite ends. Understanding how to compute these tensions is crucial when analyzing systems where ropes or cables are used to maintain equilibrium, ensuring that the forces throughout the system are balanced.

Vector Decomposition

Vector decomposition refers to the process of breaking a force into its components along defined axes, typically the horizontal and vertical. This is essential in problems where forces act at angles, as it permits the analysis of the contribution of each force in each direction, simplifying the mathematical resolution of equilibrium conditions.

Static Equilibrium

This concept involves ensuring that the net forces acting on an object cancel out, resulting in no acceleration. In a static system, the sum of all forces, both in the horizontal and vertical directions, must equal zero. This principle allows one to set up equations to solve for unknown forces or tensions, such as those in ropes or cables.

Transcript

00:01 So in this question, we are told that we have three ropes and they are tied to a small light ring.

00:07 Two of the ropes are anchored to walls at right angles and the third rope pulls away from the ring as shown in the figure.

00:16 And then we are asked to find the magnitudes of tension in the first rope and the second rope respectively.

00:24 So i think to start off this problem, it's nice to just draw three body diagram.

00:29 So let's say this represents our very light ring.

00:33 And then we've got rope number one pulling horizontally away from that ring.

00:40 So that's going to be tension number one.

00:43 Then we've got rope number two pulling vertically away from the ring.

00:46 So that will be tension number two.

00:48 And then we've got a third rope pulling down at an angle.

00:52 We're told that this has a value of 100 newtons.

00:59 And it makes an angle of 30 degrees with the horizontal.

01:03 And just in case you're wondering, the reason why it says small, very light ring is that's a hint to you that you don't need to worry about the gravitational force for this particular object.

01:16 The tension forces will be much, much bigger than the gravitational force, and so we can just exclude it.

01:22 Now, something else to notice is that this object is not moving, so it is an equilibrium.

01:30 And when you have an object in equilibrium, you can use two facts.

01:36 The first one is that the sum of the forces in the x direction will be zero, and then we'll also have that the sum of the forces in the y direction is also equal to zero.

01:45 So in order to find t1 and t2, we just need to utilize these equations...

Please give Ace some feedback