The three ropes in Figure P 5.1 are tied to a small, very...

The three ropes in Figure $\mathrm{P} 5.1$ 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?

Key Concepts

Systems of Equations

Once the forces are resolved into components and the equilibrium conditions are established, the problem often leads to a set of simultaneous equations. Solving these equations allows for determination of unknown forces or tensions in the system.

Resolving Forces into Components

When forces act at angles, it is essential to break them down into their perpendicular components, typically along the horizontal and vertical axes. This procedure simplifies the analysis by allowing the application of equilibrium conditions separately in each direction.

Vector Addition

Vector addition is the process of combining multiple force vectors to determine a net effect. By adding the components of each vector, one can ensure that the algebraic sum of forces in each coordinate direction equals zero for a system in equilibrium.

Free Body Diagram

A free body diagram is a simplified representation of an object isolated from its surroundings, showing all the forces acting on it. This diagram assists in visualizing the problem, identifying all forces, and forming the basis for writing the equilibrium conditions for the system.

Static Equilibrium

This concept refers to a state where all forces acting on a body cancel each other out, meaning the net force is zero. In problems like these, the object is at rest, so the sum of forces in all directions must balance, forming the foundation for setting up equilibrium equations.

Transcript

00:01 So in this problem, we're asked about three ropes that are connected via a ring, and we're given the tension in rope three, but we want to find the tension in rope one and rope two.

00:11 So this is what they look like.

00:13 Here's that little ring in the middle.

00:16 We have tension two going up.

00:20 Tension one connected to the wall like that.

00:24 And this one pulling down.

00:26 We're given that this is equal to 100 games.

00:30 We're also given that this angle is 30 degrees with the horizontal, and this is a right angle.

00:36 So the first thing we need to do is to define a coordinate system.

00:41 So give us x and y.

00:44 And we know for the ring to be in static equilibrium, we must have from newton's first law that the net force in the x is equal to tension 1 in the x plus tension 1 in the x, plus 10 to 3 in the x, and the force net must be equal to 0.

01:03 And it's also the same thing in the y direction.

01:17 And so here we want to break down the components, of each force.

01:21 So in the x direction, we have tension 1 is equal to negative tension 1.

01:30 And notice the sign because in our coordinate systems would be in the negative x direction.

01:35 We have tension 2 is equal to 0, because there is no horizontal component to this...

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