Problem 91 - University Physics with Modern Physics

Question

Tarzan, King of the Jungle (mass \(=70.4 \mathrm{~kg}\) ), grabs a vine of length \(14.5 \mathrm{~m}\) hanging from a tree branch. The angle of the vine was \(25.9^{\circ}\) with respect to the vertical when he grabbed it. At the lowest point of his trajectory, he picks up Jane (mass \(=43.4 \mathrm{~kg}\) ) and continues his swinging motion. What angle relative to the vertical will the vine have when Tarzan and Jane reach the highest point of their trajectory?

Step-by-Step Solution

Answer

Answer: To find the angle relative to the vertical when Tarzan and Jane reach the highest point of their trajectory, follow the provided step-by-step solution and use the given values such as the mass of Tarzan, mass of Jane, length of the vine, and initial angle. After calculating the height at the highest point of the trajectory, use trigonometry to find the final angle relative to the vertical.

Step 1: Calculate the initial total potential energy of Tarzan with respect to the lowest point of his trajectory.

At the initial position of Tarzan, we can calculate the height (\(h_{i}\)) he is above the lowest point of his trajectory using the length of the vine and the given angle: \(h_{i} = L - L \cos (\theta)\) Where \(L = 14.5\;\text{m}\) is the length of the vine and \(\theta = 25.9^\circ\) is the given angle. Now we can find the total initial potential energy: \(PE_{i} = m_{t} g h_{i}\) Where \(m_{t} = 70.4\;\text{kg}\) is the mass of Tarzan and \(g = 9.81\;\text{m/s}^2\) is the acceleration due to gravity.

Step 2: Calculate the total kinetic energy Tarzan has at the lowest point of his trajectory.

At the lowest point of his trajectory, all of Tarzan's potential energy has been converted into kinetic energy. So, the total kinetic energy will be: \(KE = PE_{i}\)

Step 3: Calculate the combined mass of Tarzan and Jane.

To find the combined mass of Tarzan and Jane, we can just sum their individual masses: \(m_{tj} = m_{t} + m_{j}\) Where \(m_{j} = 43.4\;\text{kg}\) is the mass of Jane.

Step 4: Calculate the total energy of Tarzan and Jane when they are at the lowest point of their trajectory.

At the lowest point of their trajectory, the total energy of Tarzan and Jane will be the sum of their kinetic energy and the potential energy due to the mass of Jane: \(E_{tj} = KE + m_{j}gh_{l}\) Where \(h_{l}\) is the height of Jane when she is picked up by Tarzan. Since she is picked up at the lowest point, the height will be zero: \(E_{tj} = KE\)

Step 5: Calculate the height at the highest point of their trajectory using conservation of energy.

The total energy at the highest point of their trajectory will be equal to the potential energy: \(PE_{f} = m_{tj}gh_{f}\) Using conservation of energy: \(E_{tj} = PE_{f}\) We can now solve for \(h_{f}\), the height at the highest point of their trajectory: \(h_{f} = \frac{E_{tj}}{m_{tj}g}\)

Step 6: Use trigonometry to calculate the angle relative to the vertical at the highest point of their trajectory.

To find the angle at the highest point of their trajectory, we can use the length of the vine and the height: \(\cos(\theta_{f}) = \frac{L - h_{f}}{L}\) Where \(\theta_{f}\) is the final angle relative to the vertical. Now, we can calculate the final angle by taking the inverse cosine: \(\theta_{f} = \arccos\left(\frac{L - h_{f}}{L}\right)\) Following these steps will allow us to find the angle relative to the vertical when Tarzan and Jane reach the highest point of their trajectory.

Key Concepts

Kinetic Energy

Kinetic energy is the energy an object possesses due to its motion. It's a vital concept in mechanics and is usually denoted by the symbol KE. The formula for kinetic energy of a moving object is given by
\( KE = \frac{1}{2}mv^2 \)
where m is the object's mass and v is its velocity. In the Tarzan and Jane problem, when Tarzan is at the lowest point of his swing, all his initial potential energy from being at a height is converted into kinetic energy. As there is no other energy input or dissipation, the initial potential energy and kinetic energy at the lowest point are equal in magnitude. This conservation of energy allows us to set up an equation that equates the two and solve for unknown variables.

Potential Energy

Potential energy is the stored energy an object has due to its position or state. In the context of our problem, gravitational potential energy is most relevant, which is dependent on the object's height above a reference point, its mass, and the acceleration due to gravity (g). The formula for gravitational potential energy (PE) is:
\( PE = mgh \)
where m is mass, g is acceleration due to gravity, and h is the height above the reference point. In the step-by-step solution, potential energy calculations establish the energy states at different points in Tarzan's swing. Understanding potential energy is crucial for solving problems involving height and gravity.

Trigonometry in Physics

Trigonometry, the branch of mathematics dealing with the relationships between the sides and angles of triangles, is essential in physics, particularly in mechanics. It provides the tools to model and solve problems involving angles, heights, and distances. In the Tarzan problem, trigonometry helps us determine the change in height as he swings by using the vine's length and the angle made with the vertical. The basic trigonometric functions sin, cos, and tan are often used in such calculations. Specifically, the original exercise uses the cosine function to relate the angle of the vine to the height of Tarzan above the lowest point of the swing, allowing us to calculate the initial and final potential energies.

Mechanics

Mechanics is the branch of physics dealing with the motion of objects and the forces causing those motions. It typically involves analyzing situations to determine the net effect of forces and applying conservation laws, such as conservation of energy and momentum, to predict the behavior of physical systems. The Tarzan and Jane exercise is rooted in mechanics, as it combines principles of energy conservation with the forces acting on the system. By considering the mechanics of Tarzan's swing, the combined masses, and the pivotal concepts of kinetic and potential energy, one can determine the behavior of the system at different points in time, ultimately leading to the calculation of the final angle of the vine.