# The formula for tension in a rope attached to a weight at an angle

Aug 26, 2022

## Bungee Cords

I learned on a mil. spec. system. In pirate jumping it has a number of advantages. You have a set of 5 individual cords that you group together in a set of 3,4,or5. So you can jump from 100-250lbs with one set of cords. They are covered in Nylon (or Cotton) sheaths that protect them from dirt when you lay them down. Also, the redundancy of multiple cords, as well as the sheaths which act as a static back up in the case of over elongation, provide (I believe) a greater degree of safety. Traditional mil. spec. cords have an elongation of about 110%. By the way, in the “bungy or bungee” page a common mistake is made about this word. Elongation refers to the change in length of the cord, not the ultimate loaded length. So 110% elongation in a 100 foot long cord is 110 feet for a total stretched length of 210 feet, or 2.1 times it’s original length. Since mil. spec. cord stretches less, a longer length is often used, resulting in more initial free fall.

After the 1995 X-games I realized that, because of the bigger rebounds, more acrobatic stunts were possible on all rubber cord. So I set about learning how to build that type of cord. New Zealand style cord is built with a ribbon, about 3 inches wide, composed of a number of stands of rubber. This is wrapped around teflon spools, tied together, stretched to it’s ultimate elongation and wrapped down it’s length, with the same type of rubber, to hold it all together.

The actual building of an all rubber cord, or tying loops on the ends of mil. spec. cord is not something that can be learned through words only. You need to learn first hand from someone who knows what they are doing. So you can watch them and they can check your work. The same goes for the bungee system, whether it’s a lowering system, a man basket on a winch, or a raising system, you either need to learn it first hand or start from scratch with the engineering done on all the materials, and many practice “jumps” done with sandbags.

New Zealand specification cord has an elongation of 200-300%, i.e. it will stretch to 3-4 times it’s original length. I believe the Kocklemans came up with the idea of building a length of webbing into all rubber cord. This back up is the length of the ultimate elongation of the cord. This gives it the redundancy that I like. The Kocklemans build their cords thicker than New Zealand specs. This gives the cord a long life (over 1000 jumps) but results in higher G-forces, and less ability to do rebound tricks. I believe their elongation is 150-200%.

It is difficult to determine actual elongation because many people will tell you the length of the all rubber cord before it is stretched out and tied down it’s length. This makes the cord a little longer (about 5-10%) than when the ribbon is first layed around the spools. For example, for a 100 foot jump you would first lay the rubber out at 22.5 feet, which would need to stretch to 4.5 times it’s original length to get to 100 feet. But once the cord is stretched and tied it would be about 25 feet long. So then it would need to stretch to 4 times it’s length to reach 100 feet.

## The Euro Cord

The European/New Zealand cords described above look like this

Steve who wrote that section above believes that the US mil spec system is safer because of the extra redundancy, but the Euro cords do have the advantage that they can be inspected after each jump for signs of rubber failure or possible problems. The sheath on Mil spec cords prevents this. Some Euro cords have an extra static line built into the core to prevent over stretch and provide extra backup. The term Euro cord is a little misleading – this cord is used and made in many countries.

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## Mil Spec Cord

Mil Spec cord is cord that is made to US military specifications, these specifications were designed to hold down tanks on boats and in planes rather than to suspend adrenaline junkies jumping off bridges. As it turns out they work rather well for both!.

Here is some mil. spec cord made by Glenn from Bungee Experience. The pink/white stuff is the cord, the blue webbing is attached in a very special way to the ends of the cords so that carrabiners can be attached to each end. The way the end of the cord is connected with a piece of webbing is called termination. The Red Tape is part of the special way the cords are terminated.

## Specs

Lots + lots of people have asked me for bungee cords specs and I just got emailed this from J Kockelman (cheers!)

Most Bungee Jumping Cord is made from “natural rubber” whose physical constants are in most college libraries. The most common bungee cords stretch 2 to 4 times the original length and the jumper feels 2.5 to 3.5 G’s.

I found a page on bungee.com (link expired) which had a technical paper on Bungee jumping.

## The formula for tension in a rope attached to a weight at an angle

Tension force is developed in a rope when a weight is attached to it. The tension developed in the rope should be equal to the gravitational pull on the weight. But this is true only for a case when the rope is vertically suspended. The formula for tension also depends on the angle of suspension. Here in this article, we look at the examples, formulas, and numerical problems for tension when the rope is suspended at an angle to the ceiling.

The weight of mass m is suspended by two ropes with tension T1 and T2. The tension in both the ropes will be different, so we have to derive two separate sets of the formula for tensions in both strings. Since the weight is static, the net forces acting on the weight in the x and y direction should be zero.

We will start by drawing a free-body diagram and resolve the forces in x and y directions.

Resolving the forces in y-direction: The forces acting in the y-direction are a downward gravitational pull and component of tension forces T1 and T2 in an upward direction. Equating the force we get:

T1 sin(a) + T2 sin(b) = m*g ———-(1)

Resolving the forces in x-direction: The forces acting in the x-direction are the components of tension forces T1 and T2 in opposite directions. Equating the forces we get:

T1cos(a) = T2cos(b)———————(2)

Solving equations (1) and (2), we get the formula for tension.

T1 = [T2cos(b)]/cos(a)]

T2 = [T1cos(a)]/cos(b)]

From the above equations, we can also infer that the higher the angle of suspension higher will be the tension, with the maximum tension at a suspension of 90 degrees.

## Problems on tension formula

The weight of a mass of 10 kg is suspended by two ropes at an angle, find the tension in the ropes.

We will follow the same approach as we used above for deriving the formula for tension. We will first resolve the forces in the x and y direction and form two equations.

Resolving the forces in the x-direction we have:

T1 sin30 + T2 sin60 = 98 N ———-(1)

Resolving the forces in the y-direction we have:

Solving the above equations we get the values for tension T1 and T2:

T1 = 65.1 N T2 = 112.3 N

We can see that T2 is higher than T1, which also shows that angle b is greater than a.

Caculate the tension in the string attached to a kite, provided there is no drag and the string is not pulled the boy.

Here in this case we have only a single string, so we can just resolve the forces in the y-direction and get the formula for tension.

The tension in the string is much more than the lift generated by the wind. This tension is provided by the muscular force in the hand of the boy holding it.

## Tension in a horizontal suspension

A weight of 10 Kg is suspended horizontally by two ropes with tension T1 and T2. The length of the ropes is equal.

Now, in this case, the angles formed by the ropes with the ceiling are 0. Let us try resolving the forces and find the value of tension in the ropes. But such problems can be solved more easily logically rather than by equations. Such problems can be solved only under many assumptions like the equal length of ropes, massless ropes, 100% horizontal suspension, etc.

In the x-direction:

T1 and T2 are opposite, and there is no net force acting in the x-direction, hence they must be equal. Let us apply the formula for tension and verify this.

T1cos0 = T2cos0 => T1 = T2

Now, in the Y-direction:

The entire weight of the box has held the ropes, so the sum of the tensions should be equal to the weight. Let us apply the formula for tension to find out.

## Effect of Body Mass and Cord Length on Bungee Jump Motion

This essay investigates a body’s motion during a bungee jump in order to answer the question: “To what extent a body’s mass and length of the cord affect the Bungee jumping motion?” the investigation takes place with comparing three different bungee cords ‘s performance in two simulation laboratory experiments. The first is to check the relation between the bungee jumping cord and its relation to Hooke’s law and finding its elastic limit. The second is to inspect the motion in terms of velocity and acceleration changes with varying the weight of the body attached to the cord and changing the length of the cord, since they are the factors to be considered in the research question.

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## Scope of work

The vine jumpers of Pentecost Island in Vanuatu inspired the spot of Bungee jumping, as it was viewed by way of a rite of passage to manhood. It is about jumping from a high point such as a bridge, a building or a crane attached to nylon braided, rubber shock cord. It is from a fixed structure most of the time but it is possible to do it from an object floating in the air, for example, a moving crane or a hot air balloon. It became a popular sport the last two decades in the United States of America where people do it for the sake of the excitement and adrenaline pumping sensations. Of course this sport involves a lot of risk and most of the accidents occurring are from miscalculations in the length of the elastic cord, which leads to many horrifying accident when people end up landing on the surface or the cord collapses, it occurred to my mind that exploring such a occurrence might be very interesting.

In this essay I aim to look at the physics behind Bungee Jumping. The aim of this essay is to investigate the factors affecting the bungee jump motion. I will be exploring the stages that the bungee jump goes through and the factors affecting it allowing a safe landing but exciting at the same time. This involves data logging from laboratory experiments and graphing data with analysis. Exploring this matter can easily make connections between fundamental concepts of physics and real world phenomena: Bungee Jumping. Therefore attempting to answer, “To what extent a body’s mass and length of the cord affect the Bungee jumping motion?”

## Safety of Bungee jumping

There is no doubt that a thrilling from a height usually more than forty-five meters carries its own risk and can be very dangerous, Bungee jumping is like most adrenaline pumping sports, when done wrong, can be hazardous and even lethal.

Bungee jumping mishaps can occur because of faulty equipment or regardless of safety measures, the injuries that could have been avoided are human errors when the body strapping fails due to improper attachment or flawed harness, Chris Thomas is an example of this horrible incident, he died during a charity jump in Swansea, Wales: because of his weight[1]. Another case is cord length miscalculation and the jumper ends up hitting the ground or the bungee cord just snapped, similarly to what happened to Erin Langworthy, an Australian woman who almost drowned with her feet tied together in Zambezi River at Victoria Falls[2]. In 1989, this activity was banned in France and one state in Australia after three people faced their death[3]. And many other incidents causing people to collapse on concrete and suffer from extreme cranial trauma or even die because the rope was too long, that actually happened to Matthew E. Coleman[4], who died at an Adventure World bungee jump.

However, unavoidable injuries might occur, minor injuries such as skin burn, which is triggered through gripping the cord, happen when Bungee jumpers do not act accordingly to the guidelines given. Some of them stated that they got slapped in the face by the cord.

Other mores serious injuries; such as eyesight damage or temporary retina haemorrhage[5], strokes and traumatic carotid artery dissection happened to fit and healthy youth.

But injury inflicted by the cord, such as choking to asphyxiation, appears not to happen. This can be explained by a combination of factors, including the cord’s minimal torsional stiffness. Also, the minor pendulum motion keeps the cord from contacting the jumper and tangling or strangling him,

No modern-day jump site has seen any serious entanglement, and it is noteworthy that many participants enjoy somersaulting during the free fall without any harm or disaster occurring.

## Principle components in the physics of Bungee jumping

To allow the bungee jumping motion to occur the person jumps from a high surface and the cord stretches as he is moving downwards, this demonstrates the cords’ elasticity, which can be defined as the ability of a body or the cord, in this case, to oppose a force exerted on it and change shape and size and to return to its same characteristics when the strain is removed.

The law of elasticity, Hooke’s law, determined by Robert Hooke, an English scientist in 1660, which states that, for relatively small deformations of an object, the displacement or size of the extension is directly proportional to the deforming force applied.

Under these conditions the object returns to its original shape and size upon removal of the load.

If the force exerted exceeds a certain amount, known as the elastic limit, it would create a permanent deformation to the body even when there is no force applied on the body. The elastic limit differs from a body to another because both of the resistance to stress and it depends on what the body is made of. Elastic materials expand thinner and thinner until rupturing at their breaking point.

The strength of materials is the measurement of a body’s capacity to bear strain and stress. Stress is the internal force applied by a segment of an elastic body upon the connecting part and strain is the dimension’s deformation caused by stress. Elastic materials are the materials whose stress disappears after the exerted force is removed.

In bungee jumping the cord is subjected to pull, this is identified as tension. When the cord has weight attached and it is being pushed, this is known as a compressive stress. During the jump, the external forces twist the body around an axis, it is known as the torsional stress.

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## Experiment 1: Hooke’s Law

This experiment is carried out to calculate the elasticity of the bungee cord and its elastic limit.

## Variables

• Independent variables:
• Dependent variable:
• Control variable:

The same bungee cord used for different weights

Shape of the weight used

Height of the cord from the ground

## Apparatus Used

Since the Hooke’s law experiment apparatus is usually equipped with a retort stand, which is a stand that has a ruler and a pointer attached to the spring, but since I am using a bungee cord instead of the spring, I used a regular clamp and I had seven different masses labeled 0.1 kg, a digital measuring scale with 0.01 kg uncertainty, three different car bungee cords purchased at the local hardware shop, a ruler0.0005m and a flat surface to perform the experiment on.

## Method

First of all, I measure the length of the car bungee cord is provided with two hooks at each of the extremities, therefore I hang the cord with one hook on the clamp, I measure the weight holder then I hang it to the bottom hook line the I add one weight cylinder, afterwards I carefully measure the length of the cord. Next I measure each weight on the scale and I measure the extension on the cord each time the weight is added. All the measurements are recorded during the experiment.

## Variables

• Independent variables

Length of the cord

Thickness of the cord

• Dependent variable:

Time taken to complete a bungee jump

Velocity of the body

• Control variable:

The same bungee cord used for different weights

Shape of the weight used, using the same set of weights

Height of the weights from the motion sensor, it is controlled by placing the Vernier motion sensor on a laboratory chair with the ability to move it around and adjust its height.

## Apparatus Used

Vernier motion sensor connected to a computer with a data logging software[6] installed which will be crucial for more accurate timing and graphing purposes than manual timing. Also, the same Bungee cords used in the previous Hooke’s law experiment are used since characteristics are already measured and this experiment is relating to the previous one, a meter ruler can be used to insure that the apparatus is perpendicular to the motion sensor. Blu-tack and tape is also needed.

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## Method

A clamp is put on a flat surface, to prevent it from falling I and the bungee cord is hung from it with the weight holder suspended at the bottom of the cord, both of the hooks attached to both ends of the cord are secured with Blu-Tack to prevent the apparatus from falling and act as the harness in this simulation. The Vernier motion sensor is put on the laboratory chair, and before starting the experiment, activate the motion sensor and oscillate the cord with the suspended weight holder to test the sensitivity of data logging and test the range of motion detection. Afterwards the weight is elevated to the beginning of the cord and it is released with minimum to no force. This step is repeated by adding weights and the weights are secured with a thin strip of tape to avoid them falling off.

## Bibliography

“Aussie Plunges into Raging Waters after Bungy Cord Snaps.” N.p., 9 Jan. 2012. Web.

“BERSA.” Bersa. N.p., n.d. Web. 19 Nov. 2014.

“Bungee Jumping.” Wikipedia. Wikimedia Foundation, 18 Nov. 2014. Web. 18 Nov. 2014.

“Elasticity.” The Columbia Encyclopedia. 6th ed. N.p.: Columbia UP, 2014. Print.

“Fatal Bungee Jump Was “accident”” BBC News. BBC, 25 Feb. 2005. Web. 19 Nov. 2014.

“For Thrills, Lovers and Others Leap.” The New York Times. The New York Times, 30 July 1991. Web. 19 Nov. 2014.

“Hooke’s Law.” Encyclopedia Britannica. N.p., n.d. Web. 19 Nov. 2013.

“Hooke’s Law” The Columbia Encyclopedia. 6th ed. N.p.: Columbia UP, 2012. Print.

“Injuries Resulting from Bungee-cord Jumping.” ANNALS OF EMERGENCY MEDICINE 22.6 (1993): 1060-063. Print.

“PhysicsLAB: Springs: Hooke’s Law.” PhysicsLAB: Springs: Hooke’s Law. N.p., n.d. Web. 18 Nov. 2014.

“Relatives Grieve after Fatal Bungee Accident.” Baltimore Sun. N.p., 16 May 2000. Web. 19 Nov. 2014.

“Strength of Materials.” The Columbia Encyclopedia. 6th ed. N.p.: Columbia UP, 2012. Print.

[1] “Fatal Bungee Jump was “accident” ” BBC News. BBC, 25 Feb. 2005. Web. 19 Nov. 2014.

[2] “Aussie Plunges into Raging Waters after Bungy Cord Snaps.” N.p., 9 Jan. 2012.Web

[3] “For Thrills, Lovers and Others Leap.” The New York Times. The New York Times, 30 July 1991. Web. 19 Nov. 2014.

[4] “Relatives Grieve after Fatal Bungee Accident.” Baltimore Sun. N.p., 16 May 2000. Web. 19 Nov. 2014.

[5] “Injuries Resulting from Bungee-cord Jumping.” ANNALS OF EMERGENCY MEDICINE 22.6 (1993): 1060-063. Print.

[6] Logger Pro 3. Portland, Or.: Vernier Software, 2003. Computer software.

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