Three people located at A, B, and C pull on ropes tied to a ring. Find the magnitude and directi

Three people located at A, B, and C pull on ropes tied to a...

Key Concepts

Resolution of Forces into Components

This technique involves breaking a force into perpendicular components, typically along horizontal and vertical axes. By resolving forces into components, complex vector equations simplify into sets of scalar equations, making it easier to apply and satisfy the equilibrium conditions in each direction.

Free Body Diagram

A free body diagram is a simplified representation of a body isolated from its environment, depicting all external forces acting upon it. It is an essential tool in statics and mechanics for visualizing forces, setting up equations based on equilibrium conditions, and ultimately solving for unknown forces or tensions.

Vector Addition

Vector addition is the process of combining forces taking into account both their magnitudes and directions. In equilibrium problems, vector addition ensures that the sum of all forces in different directions yields a zero net force. This process can be performed graphically using the head-to-tail method or analytically by resolving forces into their components.

Equilibrium of Forces

This concept states that a system is in equilibrium when the net force acting on it is zero. In statics, this means that all the individual force vectors, when added together, cancel each other out. It is a fundamental principle used to determine the conditions under which a body remains at rest or moves without acceleration.

Transcript

00:02 All right, we're given a question that has three people pulling on a rope, pulling on three different ropes.

00:16 This is vector a equal to 100 pounds of force.

00:23 This is a 45 degree angle.

00:28 And we have a vector going in this direction.

00:32 This is a 30 degree angle.

00:36 And our force 2 is 60 pounds.

00:43 And then we have another person here.

00:47 Person c that is pulling with force f3.

00:54 We want to determine the angle and the direction that it appears person c must pull so that no one moves so that the system is at equilibrium.

01:12 So the first thing is for us to find our force vectors.

01:24 And figure out our f1 in terms of our x and y, and then figure out our force 2 in terms of our force x and y.

01:39 And then if we combine those together in the negative direction, this would be the positive direction of our forces, of our forces 1 and forces 2 combined.

01:53 If we combine them and put them in the negative direction, we could calculate for a magnitude of force three.

02:01 So let's go ahead and break this down.

02:03 We'll take force one, start with force one.

02:08 And what we're going to do is we're going to get the x and the y force components.

02:18 Okay, so we're going to say, well, f1 is the same thing as 100 multiplied by.

02:30 By cosine of this whole angle here, which would be 180 minus 45, which is 135 degrees.

02:52 And then we have our force in the y direction, which will be the same thing, but sign 135.

03:08 Okay.

03:09 And this will come out to be negative 50, square root two.

03:18 And we'll have a 50 square root 2.

03:26 We're going to repeat this now for force 2.

03:31 So it's going to be 60 cosine 30 degrees, 60 sine 30 degrees.

03:46 And that will come out 30 root 30.

03:57 I'm sorry, 30 root 3.

03:59 And finally, okay, now we want to find our force 3, which will be the same thing.

04:10 As negative force 1 plus force 2 we'll say that's the same thing as we'll pull this back here it's going to be equal to if we add our signs together all of this will be negative excuse me if we add all of our cosine values together it'll be negative 50 square root 3 plus 30 square root 3...

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