(II) The two trees in Fig. 9-51 are 6.6 m apart. A backpacker is trying to lift his pack out of

step 1 Once again, welcome to a new problem. This time we have two trees. One is on the left like that.

(II) The two trees in Fig. 9-51 are 6.6 m apart. A...

Key Concepts

Small Angle Approximation

When the sag of a rope is small relative to its span, the angles formed by the rope with the horizontal are typically very small. In such cases, the small angle approximation (where sin(?) ? ? in radians, and tan(?) ? ?) can simplify the trigonometric relationships, making it easier to solve for unknown forces without significant loss of accuracy.

Statics Equilibrium

In problems involving suspended objects or systems at rest, the concept of statics equilibrium is fundamental. It requires that the sum of all forces (and torques) acting on the system be zero. In this context, the backpack, the forces from the rope tension, and the externally applied force must balance, meaning that both the horizontal and vertical components of the forces cancel out.

Tension in Ropes

The tension in the rope is the force transmitted along the rope when it is pulled taut by forces acting at its ends. It is important to understand how this tension decomposes into vertical and horizontal components, especially when the rope sags, because these components are used to counteract the weight of the backpack and any additional force applied.

Force Decomposition

Decomposing a force into its horizontal and vertical components is a key concept in mechanics. This process involves using trigonometric functions to resolve a force vector into components that act along mutually perpendicular directions. In this problem, the tension in the rope is split into components that balance the weight of the backpack and the applied downward force at the midpoint.

Geometric Analysis of Sag

The sag of a rope suspended between two supports introduces a geometric relationship between the sag height, the span of the rope, and the angles that the rope makes with the horizontal. By analyzing the geometry of the rope’s shape (often approximated by simple relations for small sags), one can determine the angles needed to relate the applied forces to the rope tension.

Transcript

00:01 Once again, welcome to a new problem.

00:05 This time we have two trees.

00:08 One is on the left like that.

00:12 And then, well, i think i need to make the trees really good.

00:19 So, you know, one tree is on the left.

00:24 And the other one is on the right.

00:28 So we have two trees.

00:33 And then in between the trees, there is, hanging a street light or street of street a street lamp okay so it's a so we have a backpacker and is trying to lift his pack out of the reach of bears so this is this is the backpacker over here you know trying to pull down it's like a rope it's going like that at an angle like that and this is hanging so the backpacker is a bag right here hanging and and the backpacker is pulling down with the force f is pulling down with the force there the piece of information that we're given is the fact that in between the the two trees happen to be 6 .6 meters apart.

02:05 Okay, 6 .6 meters apart.

02:10 The backpack has a mass of m equals to 19 kilograms.

02:24 And then, so we want to find, so this is the information that's given and we want to find.

02:38 The force.

02:39 We want to find the force f.

02:45 So the force that must hold the backpack such that, you know, a, the rope sucks at the midpoint by 1 .5 meters.

03:17 That's part a.

03:19 And then part b, we're saying the rope is sagging by 0 .15 meters.

03:25 So that's the information that we're given.

03:31 I want to find, you know, what's this force, what's this required force that makes the rope suck by either 1 .5 meters in the middle or 0 .15 meters.

03:51 So, you know, the first thing we're going to do is we'll have like a free body diagram and the system is in equilibrium system is in equilibrium therefore the sum of f x equals to zero meaning that forces on the x in both directions assuming you know this could be the positive x and this negative x they're in equilibrium so it's zero and then the sum of f of y is also to zero meaning that the forces up and down are in equilibrium so they cancel each other out the free body diagram looks like this you can see the force there is a force right there and then the distance we're looking for is right here which is you know it's the wide distance and then the total the total distance across.

05:04 It's the length is l.

05:06 And so half of it will be l over two.

05:11 You know, that's half of the distance.

05:13 There's an angle theta.

05:15 That's this angle right here that it makes with the system data.

05:21 And then their forces acting on the system.

05:25 So, you know, on one side we have this is the actual force.

05:33 That's pulling upwards, this force right here.

05:37 We're calling it f, and that's what we want to calculate.

05:40 But f has two components.

05:42 One of them is f cosine of theta, and then the other one is f sine of theta.

05:53 F sine theta.

05:58 And then also on this side, we have still f, and on top of the previous f, sign, there's another f -sign theta component pushing upwards and this one is pulling to the right.

06:18 That's f -cosine of theta.

06:21 And you can see the system is in equilibrium.

06:24 So along the x -axis, this is what happens and then along the y -axis, that's what happens.

06:30 For the y -axis to be in equilibrium, it means that the weight of the backpack is pushing downwards like that or pulling down ons like that the entire distance the entire distance of the system is l you know so this is this is the information i was seeing right now if we think about data itself would say you know and i can i can follow that up in the next page would say that would say that tangent of theta tangent of theta is the opposite, which is y, over the adjacent, which is l over 2.

07:22 Remember, if you look at the diagram, you get to see that if this angle is theta, then the opposite of theta is y, and then the adjacent of theta is l over 2.

07:36 So coming back, we see that i can solve for theta by getting the turn inverse of theta, which happens to be y over l over 2.

07:47 That's the angle itself.

07:50 That's the theta angle...

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