Before the trunk of a large tree is felled, cables A B and B...

Key Concepts

Statics Equilibrium

Statics is the study of bodies at rest or in constant motion. In this context, ensuring that the computed forces and moments are balanced assures that the structure remains in equilibrium, an essential consideration in many engineering applications such as tree felling and cable support systems.

Moment Arm

The moment arm is the shortest distance from the pivot point to the line along which the force acts. This concept is crucial in calculating the moment since only the component of the force perpendicular to this distance contributes effectively to rotation.

Vector Addition

Vector addition is the method used to combine multiple forces or vectors. Each force is broken down into its components, which are then summed to yield the overall resultant, facilitating the analysis of forces acting in different directions.

Resultant Force

This involves combining individual forces (in this case, forces from the cables) into one net force, both in magnitude and direction. The process requires vector addition, which considers each force's direction and magnitude, ensuring the net effect is correctly captured.

Moment (Torque)

A moment, or torque, measures the tendency of a force to cause rotation about a pivot or a point. It is calculated as the product of the force magnitude and the perpendicular distance (moment arm) from the point of rotation to the line of action of the force, encapsulating both the magnitude and leverage of the force.

Transcript

00:01 Hello.

00:02 This problem asks us to find the moment that a tree trunk will experience when attached to two cables, cables ab and cable c .b.

00:12 Cable a .b has a tension of 555 newtons.

00:16 Cable cb has the tension of 660 newtons.

00:20 Now the steps that we're going to take to solve this problem is first we're going to find all over ours.

00:25 These are going to be our vector descriptions of each cable.

00:28 Next, we're going to go ahead and find their magnitudes.

00:33 Using this new information that we've found, we're going to then find our newton's per meter description.

00:40 That's going to tell us how much force is being applied to each meter of cable.

00:46 And then finally, we're going to solve the problem.

00:49 All right, step one.

00:51 First, we need to find our descriptions of both cables.

00:55 Let's go ahead and start the first cable.

00:57 So i'm going to write down r, a, b.

01:00 Now, first we need our x component.

01:03 Our x component is negative 0 .75.

01:09 And then we need our y component.

01:12 Now remember, we're going to get the aby component by subtracting the y value that b has from the y value that a has.

01:25 That's going to give us 0 minus 7.

01:31 So that's going to leave us with a negative 7.

01:37 And now we need our z component, which is going to be 6.

01:41 So plus 6.

01:46 Now because these are descriptions of distance, we need to add a label, and we're going to have meters.

01:53 Now let's find the description of our next cable.

01:59 Again, we need our x value.

02:02 Our x value is going to be 4 .25.

02:07 Our y again is negative 7 and then our z is going to be simply 1 great now that we have these vector descriptions of each cable we can now go on to find their magnitude now start in the proper order again so we're going to need so we're going to find our magnitude of cable ab this magnitude is going to tell us how long the cable ab actually is.

02:48 So in order to do that, we need to first square each component, add them together, and then take the square root.

02:56 So we're going to get negative 0 .75 squared plus negative 7 squared plus 6 squared.

03:14 And then we take the square root of that.

03:18 If we put these in our calculators, we're going to find that our magnitude is 9 .25 meters.

03:24 Again, this is the distance of cable ab.

03:29 Now, let's do it for our other cable, cb.

03:36 So we're going to have 4 .25 squared plus negative 7 squared plus 1 squared.

03:55 Plug that into our calculators, and we get 8 .25.

04:03 Now that we know this, we can now determine how much force is being applied per unit length in each cable.

04:15 In order to do that, i'm going to go ahead and label this as t...

Please give Ace some feedback