A tugboat tows a barge at constant speed with a 3500 -kg cable, as shown in FIGURE 6-66 . If the

A tugboat tows a barge at constant speed with a 3500 -kg...

Key Concepts

Vector Decomposition

Vector decomposition involves breaking down a force into its horizontal and vertical components. This is crucial in problems where forces are applied at angles, allowing one to isolate the component responsible for forward motion from the overall force.

Newton's Laws of Motion

Newton's laws, particularly the concept of equilibrium in constant velocity motion, are essential. When an object moves at constant speed, the net force acting on it is zero. This principle guides the analysis of forces involved in towing, ensuring that all force components balance appropriately.

Tension in a Cable

Tension is the pulling force transmitted along a cable, rope, or chain when forces act at its ends. In towing and similar applications, tension must overcome both the weight of the cable and any opposing forces, and understanding how it varies along the cable is key to determining the effective force transmitted.

Trigonometric Functions in Force Resolution

Trigonometric functions such as sine and cosine are used to relate the components of a force vector to the angle at which the force is applied. These functions provide the mathematical basis for calculating the horizontal (forward) component of a force when its magnitude and angle are known.

Transcript

00:01 Alright, question 66 states that a tugboat tows a barge at constant speed with a 3 ,500 kilogram cable, as shown in this figure here.

00:09 Our tugboat is in the front.

00:11 Our barge is in the back, and naturally there be some sloping down of the cable, as it goes from the tugboat to the barge.

00:21 If the angle the cable makes with the horizontal, where it attaches to the barge and the tugboat is 22 degrees, find the force the cable exerts on the barge in the forward direction.

00:32 So since the tugbo's on the left -hand side, i'm going to call my positive x going to the left, and i'll just choose to be my positive y going up.

00:41 So it's consistent with what the question asks.

00:44 So because, i mean, there's a sloping down and slipping up of the cable naturally, it's easier just to simplify the situation shown in the bottom where if we have our tugboat in the front and our barge in back, we can just take the cable to be bent and somewhere in the middle, actually.

01:01 And the angle it makes with the water on the bottom here is this data in both scenarios.

01:06 So this is the situation we're dealing with.

01:10 We want to find the force the cable exerts on the barge.

01:14 So the force at this point on the barge based on the cable.

01:25 And so if you look at this scenario, it's probably easiest to deal with it here.

01:32 So if we look at our barge, actually i'll do the next page.

01:43 On the barge, if you look at our forces in our x direction, we know that the cable will be pulling to the right, and if our tension is pulling down, and this is angle theta, then our horizontal component would be the cosine, shown here, a component of the tension.

02:05 We say net force in our x direction, because initially i defined x to be positive to the left.

02:14 Note is just t -cose -theta, which will equal because it is moving to the positive direction, and as a question states, the force, i'm going to call force acting, i'll call it force barred, fb, the force acting at the barge.

02:37 And again, this is what we're solving for fb, the force acting of the barge.

02:43 We don't know attention, so we can't fully answer this question.

02:46 So this is a hint to us that we will need to look at the force in the y component of this.

02:54 So looking at the y, it's best to be best to look at this in the center to see that the y forces acting on it.

03:01 You know, i see there's a tension force here in the middle and tension force going up.

03:05 They're equivalent.

03:07 So there'll be two sources of forces in the wide component going up.

03:12 So we have tension here.

03:14 Our vertical component of the tension is given by sign.

03:19 So because there's two, one on each side, they have the balance to form...

Please give Ace some feedback