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

[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

 

[PhanthomJay]

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.

 

[kyle8921]

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.