A mountain climber, in the process of crossing between two...

A mountain climber, in the process of crossing between two cliffs by a rope, pauses to rest. She weighs $535 \mathrm{~N}$. As the drawing shows, she is closer to the left cliff than to the right cliff, with the result that the tensions in the left and right sides of the rope are not the same. Find the tensions in the rope to the left and to the right of the mountain climber.

Key Concepts

Moments (Torque)

Moments, or torque, refer to the turning effect produced by a force acting at a distance from a pivot point. In scenarios where forces are not evenly distributed, analyzing the moments allows one to set up equations about rotational equilibrium, which are vital for determining the relationships between unequal tension forces.

Free-Body Diagram

A free-body diagram is a simplified representation of an object isolated from its surroundings, showing all external forces acting on it. This tool is crucial in solving equilibrium problems as it helps to visually identify and balance all forces—such as the climber’s weight and the tensions from the rope segments.

Static Equilibrium

This concept involves an object or system that is at rest such that the net force and net moment (or torque) acting on it are zero. It is essential in problems like this where different forces, including the weight and varying tensions in the rope segments, balance out to maintain a constant position.

Tension Forces

Tension is the force transmitted through a rope, chain, or similar object when it is pulled tight by forces acting from opposite ends. In this context, the rope segments on either side of the climber experience different tension forces due to her position not being symmetrically located between the two cliffs, requiring an analysis of these forces separately.

Transcript

00:01 To solve this question, we have to calculate the magnitudes of the tensions t1 and t2.

00:05 For that, we have to use newton's second law.

00:08 And i will use the following reference frame.

00:11 A vertical axis pointing upwards and a horizontal axis pointing to the right.

00:17 Let me call this x and the vertical axis y.

00:21 Then we have to apply newton's second law to the person.

00:26 By doing that, we get the following.

00:28 For the x -axis, the net force acting on the person is equal to the mass of the person times the acceleration of the person in the x -direction.

00:37 The person is resting there, so it's not moving and not going to move.

00:41 Then the acceleration is equal to zero.

00:44 So the net force in the x -direction is equal to zero.

00:47 But the net force in the x -direction is composed by two forces.

00:51 The x component of the tension number two, so t -2 component x, and the x component of the tension number one, t1x.

01:05 Notice that t1x points to the right, so it's positive, while t2x points to the left, so it's negative in my reference frame, and this is zero.

01:17 Now, we can find a relation between t2 and t2x, and t1 and t1x.

01:24 For that, all you have to do is use the triangles that the question is giving us.

01:30 One triangle is this one.

01:34 So this is a 90 degree angle and the other triangle is this one where this is also a 90 degree angle.

01:41 With these triangles we can calculate the missing angle, this one and this one.

01:48 It's easy to see that the angles inside the triangle should sum 180 degrees.

01:55 Therefore 65 degrees plus this angle must be 90 degrees.

01:59 And then you conclude that this angle is an angle of 25 degrees.

02:08 And also doing an analogous calculation, you find that this angle is a 10 degrees angle.

02:15 Now you can decompose the tensions 2 and 1 to discover what is the relation between their x components and their magnitudes.

02:25 We have to use other triangle for that.

02:28 We use a triangle formed by the force and their components.

02:34 So for t2, it's the following.

02:37 It's a triangle like this, where the hypotenuse is the magnitude t2.

02:42 This side is t2 component x.

02:44 This side is t2 component y.

02:47 And we know that the angle between t2 and t2 x is 25 degrees...

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