Swinging on a rope problem

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In summary, the hiker needs to have a minimum horizontal speed of at least 1.8 m/s when starting to swing in order to successfully drop onto the far edge of the ravine. The equation used to solve for this speed is 1/2mv^2 = mgh, where L is the length of the rope and x is the distance from the starting point to the far edge. The correct angle to use in this equation is arcsin(x/L), not arctan(x/L) as initially thought. However, it may be simpler to find h using Pythagoras' theorem instead of finding the angle.
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Homework Statement


A hiker plans to swing on a rope across a ravine in the mountains, as illustrated in the figure, where L = 4.0 m and x = 1.8 m, and to drop when she is just above the far edge. At what minimum horizontal speed should she be moving when she starts to swing(in m/s)?

TNDhEC6.gif


Homework Equations



Ei= Ef
So...
Ke = Pe
So...
1/2mv2=mgh

The Attempt at a Solution



I have the equation set up correctly but I just don't know how to find h so I can find V

1/2mv2=mgh

The Rope makes a isosceles triangle shape so I thought of L as adjacent, and X as the Opposite so I could solve for theta by doing Θ = arctan(1.8/4.0) = 24.22º. However, upon looking at the answer online- it says to take the arcsin(1.8/4.0) = 26.7º. Why is this?
 
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##L## is not the side adjacent in the right triangle.
 

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  • #3
For that matter, x is not the base of an isosceles triangle, as TSny's diagram illustrates.
That is not the formula for finding the apex angle of an isosceles triangle anyway.
As TSny says: ##x\neq L\tan\theta## either.

I'm kinda puzzled that they want you to find the angle at all.
Since you know L and x, why not find h by pythagoras?
 

1. How does the length of the rope affect the swinging motion?

The length of the rope affects the swinging motion by influencing the period of the pendulum. A longer rope will result in a longer period, meaning it will take more time for the pendulum to complete one full swing. This can also affect the amplitude and velocity of the swing.

2. What factors can affect the stability of the swing?

There are several factors that can affect the stability of the swing, including the length and weight of the rope, the angle at which the rope is attached to the support point, and the weight and position of the person on the swing. Wind and other external forces can also impact the stability of the swing.

3. How does the angle of release impact the swing?

The angle of release can impact the swing by affecting the initial velocity and direction of the pendulum. A smaller angle of release will result in a smaller initial velocity, while a larger angle of release will result in a larger initial velocity. This can also affect the amplitude and period of the swing.

4. What is the relationship between the length of the rope and the time it takes for the pendulum to complete one full swing?

The relationship between the length of the rope and the time it takes for the pendulum to complete one full swing is directly proportional. This means that as the length of the rope increases, the time it takes for the pendulum to complete one full swing also increases. Similarly, as the length of the rope decreases, the time for one full swing decreases.

5. How does the weight of the person on the swing affect the swinging motion?

The weight of the person on the swing can affect the swinging motion by changing the center of mass of the pendulum. A heavier person will have a lower center of mass, resulting in a slower swing and a longer period. A lighter person will have a higher center of mass, resulting in a faster swing and a shorter period.

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