# Potential Energy: Suspended Person, Force of mg(3cosθ-2cosβ)

• mirandasatterley
In summary: Set those two equations equal and solve for beta.In summary, a person with mass m, suspended by two parallel ropes of length l, begins on a platform in the air and steps off with the rope at an angle beta with respect to the vertical. Air resistance is negligible. The person must exert a force of mg(3cos(theta) - 2cos(beta)) when the ropes make an angle theta with the vertical. To find the angle beta where the force needed to hang on at the bottom of the swing is twice the person's weight, you must use the conservation of mechanical energy principle and set the equations for the required force equal to 2mg. This will allow you to solve for beta.
mirandasatterley
A person with mass,m, is suspended by 2 ropes (paralell) both of length l. The person begins on a platform in the air. The person steps off the platform, starting from rest with the rope at an angle beta, with respect the the verticle. Air resistance is negligable.
a) When the ropes make an angle theta with the verticle, the person must exert a force of: mg(3 cos(theta) -2 cos(beta)).
b) Find the angle beta where the force needed to hang on at the bottom of the swing is twice the persons weight.

Any help is appreciated. Thanks

mirandasatterley said:
A person with mass,m, is suspended by 2 ropes (paralell) both of length l. The person begins on a platform in the air. The person steps off the platform, starting from rest with the rope at an angle beta, with respect the the verticle. Air resistance is negligable.
a) When the ropes make an angle theta with the verticle, the person must exert a force of: mg(3 cos(theta) -2 cos(beta)).
b) Find the angle beta where the force needed to hang on at the bottom of the swing is twice the persons weight.

Any help is appreciated. Thanks
You've got to use conservation of mechanical energy principle whereby initial PE + initial KE = final PE + Final KE. Initial KE is 0, so that's a help. For initial PE, you need to draw a diagram and see how the beta angle and radius of the arc detrmine the height h. It's a bit messy but you've got to plug and chug through it. Same for final PE at angle theta. Final KE you are trying to solve, solve for V as function of R,m,theta , and beta. Then do a FBD at the angle theta, T-mgcos theta = mv^2/r. Solve for T, the required force the person must exert.
Part b is similar, except this time you know that T = 2mg.

I would like to provide a response to the above content by first explaining the concept of potential energy and how it relates to the situation described. Potential energy is the energy that an object possesses due to its position or condition. In this case, the person suspended on the ropes has potential energy due to their position above the ground.

Now, let's break down the given equation: mg(3 cos(theta) -2 cos(beta)). The term "mg" represents the weight of the person, which is the force exerted on the person by the Earth's gravitational pull. The first part of the equation, 3 cos(theta), represents the vertical component of the force exerted by the ropes on the person. The second part, 2 cos(beta), represents the horizontal component of the force exerted by the ropes on the person. This equation takes into account the angles at which the ropes are attached to the person and the platform.

a) When the ropes make an angle theta with the vertical, the person must exert a force equal to their weight (mg) multiplied by the difference between the vertical components of the force exerted by the ropes (3 cos(theta)) and the horizontal components (2 cos(beta)). This equation shows that the force needed to hold the person in place increases as the angle theta increases, meaning that the person needs to exert more force to maintain their position as the ropes become more vertical.

b) To find the angle beta where the force needed to hang on at the bottom of the swing is twice the person's weight, we can set the equation equal to 2mg and solve for beta. This would give us the angle at which the horizontal component of the force exerted by the ropes is equal to the weight of the person. However, it's important to note that this is an idealized scenario and does not take into account factors such as air resistance or the person's weight distribution on the ropes. In reality, the angle beta may vary depending on these factors.

In conclusion, the given equation represents the potential energy of a person suspended on two ropes. It shows how the force needed to maintain their position changes as the angles of the ropes change. The angle beta where the force needed to hang on is twice the person's weight can be calculated, but it may not accurately reflect the actual situation due to other factors at play.

## 1. What is potential energy?

Potential energy is the stored energy an object has due to its position or state. It can be converted into other forms of energy, such as kinetic energy, when the object is in motion.

## 2. How is potential energy calculated for a suspended person?

Potential energy for a suspended person can be calculated using the formula PE = mgh, where m is the mass of the person, g is the acceleration due to gravity, and h is the height of the person above the ground.

## 3. What is the force of mg(3cosθ-2cosβ) in the potential energy equation?

The force of mg(3cosθ-2cosβ) represents the weight of the suspended person, which is equal to their mass (m) multiplied by the acceleration due to gravity (g) and multiplied by the trigonometric functions of the angles θ and β, which represent the orientation of the person's body.

## 4. How does the angle θ affect the potential energy of a suspended person?

The angle θ affects the potential energy of a suspended person by changing the direction of the force of gravity, which in turn affects the potential energy. As the angle θ increases, the potential energy also increases.

## 5. How is potential energy affected by changes in height and mass for a suspended person?

For a suspended person, potential energy is directly proportional to both height and mass. This means that as the height or mass of the person increases, so does their potential energy. Conversely, if the height or mass decreases, the potential energy also decreases.

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