As part of a fundraiser, the Chancellor has agreed to bungee jump from a crane 45 m above a pool filled with Jello. The plan is for the bungee cord to stop the Chancellor just before his head enters the Jello. Your task is to select a bungee cord that will safely stop the Chancellor's descent in time. To estimate the feasibility of the plan, assume for now that a bungee cord is massless and behaves like an ideal spring. Also neglect air resistance. What length and spring constant should the bungee cord have? Are those values realistic?
Things to consider:
Realistically, how much can a bungee cord stretch?
Does the bungee cord begin to stretch as soon as the Chancellor is dropped from the crane?
When is the Chancellor subject to the maximum force? How much force is safe?
To determine the length and spring constant of the bungee cord, we need to consider the Chancellor's maximum safe acceleration and the distance required for the bungee cord to stop the descent.
First, let's calculate the minimum length the bungee cord should have. The Chancellor needs to stop just before his head enters the Jello, which is 45 m below the crane. We want to avoid any risk of him hitting the Jello, so the bungee cord should be a bit longer than 45 m.
Next, let's consider the Chancellor's maximum safe acceleration. When the bungee cord stops the descent, the Chancellor experiences maximum deceleration. To ensure safety, the maximum deceleration should be below a certain limit that the Chancellor's body can withstand. This limit is usually in the range of 4 to 6 g (gravitational acceleration).
The bungee cord starts stretching as soon as the Chancellor is dropped from the crane, so we need to consider that the cord lengthens during the descent. When the Chancellor reaches the bottommost point of the descent, the bungee cord will be fully stretched.
To calculate the spring constant of the bungee cord, we can use Hooke's Law:
F = kx
Where F is the force exerted by the bungee cord, k is the spring constant, and x is the displacement (stretch) of the bungee cord.
To determine if the chosen values are realistic, we need to ensure that the maximum force exerted by the bungee cord is within safe limits. This maximum force occurs when the bungee cord is fully stretched.
Taking all these factors into account, it is essential to consult with experts who have experience in designing safe bungee cords before finalizing the length and spring constant values. They will be able to consider various factors like the Chancellor's weight and body type, the material properties of the bungee cord, and safety standards and regulations.
Remember, it is crucial to prioritize safety over other factors in any bungee jumping activity.
To select a bungee cord that will safely stop the Chancellor's descent, we need to consider the following factors:
1. Realistic stretching capability: Bungee cords typically have a maximum stretch limit to ensure safety. A general rule of thumb is that a bungee cord can stretch up to 1.5 times its original length without breaking.
2. Timing of bungee cord stretch: The bungee cord should start stretching only after the Chancellor is dropped from the crane, and not before. This is crucial to ensure that the maximum force is applied to stop the descent at the right moment.
3. Maximum force and safety: The Chancellor should experience a safe maximum force when the bungee cord is at its highest stretch. The exact force depends on various factors, including the mass of the Chancellor and the desired level of safety. It is important to consult safety guidelines and professionals to determine the maximum force the Chancellor can safely withstand.
Now, let's calculate the length and spring constant that the bungee cord should have for this scenario.
First, we need to determine the maximum allowable extension of the bungee cord. Assuming a maximum stretch limit of 1.5 times the cord's original length, we can calculate the maximum allowable extension:
Maximum extension = 1.5 * original length
Next, we need to estimate the maximum force required to stop the Chancellor's descent. This force will be influenced by the Chancellor's mass and the deceleration rate desired. Assuming a desired deceleration rate of g/3 (where g is the acceleration due to gravity), we can calculate the maximum force using the equation:
Maximum force = mass * (g/3)
Once we have the maximum allowable extension and the maximum force, we can determine the spring constant (k) of the bungee cord using Hooke's Law:
k = Maximum force / Maximum extension
To determine if these values are realistic, we would need to compare them to real-world bungee cord specifications and safety guidelines. It is recommended to consult with professionals and consider safety standards to ensure the feasibility and safety of the plan.