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Magazine Chapter 8: Problem 27
Tarzan, who weighs \(688 \mathrm{~N}\), swings from a cliff at the end of a vine \(18 \mathrm{~m}\) long (Fig. 8-40). From the top of the cliff to the bottom of the swing, he descends by \(3.2 \mathrm{~m}\). The vine will break if the force on it exceeds \(950 \mathrm{~N}\). (a) Does the vine break? (b) If no, what is the greatest force on it during the swing? If yes, at what angle with the vertical does it break?
Short Answer
Expert verified
The vine does not break. The maximum force on it during the swing is 902 N.
Step by step solution
01
Understand the Problem
Tarzan swings from a vine attached to a cliff. We need to determine if the vine breaks by comparing the maximum forces exerted during the swing against the vine's breaking strength and find out the force and angle at which it breaks if applicable.
02
Determine Energy Conversion
Tarzan's gravitational potential energy at the top converts into kinetic energy at the bottom of the swing. Use the equation: \ U_i = K_f \ to find the kinetic energy at the bottom. Here, \( U_i = mgh \), where \( h = 3.2 \text{ m} \). Since \( W = 688 \text{ N} \), we have \( m = \frac{W}{g} = \frac{688}{9.8} \).
03
Calculate Potential Energy at Top
The potential energy at the top, \( U_i \), is calculated as: \ U_i = 688 \times 3.2. \ Hence, \( U_i = 2201.6 \, \text{J} \).
04
Find Kinetic Energy at Bottom
Since all the potential energy converts to kinetic energy at the bottom, we have \( K_f = 2201.6 \, \text{J} \).
05
Compute Velocity at the Lowest Point
Use the kinetic energy formula \( K = \frac{1}{2}mv^2 \) to find velocity. Rearrange to \( v = \sqrt{\frac{2K}{m}} \). Substitute \( K = 2201.6 \) and \( m = \frac{688}{9.8} \) to find \( v \).
06
Apply Force Equation to Find Tension
At the lowest point, tension \( T \) is due to centripetal force and weight: \ T = mg + \frac{mv^2}{L}, \ where \( L = 18 \text{ m} \). Substitute \( v \) found in Step 5, \( m = \frac{688}{9.8} \), and \( g = 9.8 \).
07
Determine If Vine Breaks
Calculate the maximum tension using the values. Compare it with the breaking force of \( 950 \text{ N} \). If maximum tension exceeds \( 950 \text{ N} \), the vine breaks.
08
Calculate Angle If Vine Breaks
If maximum tension exceeds the breaking force, calculate the angle \( \theta \) where it breaks using the component of tension: \ T = mg \cos(\theta) + \frac{mv^2}{L}. \ Set \( T = 950 \text{ N} \) and solve for \( \theta \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Energy Conversion
Imagine Tarzan at the top of the cliff. He possesses a certain amount of gravitational potential energy because of his height above the ground. When he swings downward, this potential energy transforms into kinetic energy, which is the energy of motion.
At the highest point, all energy is potential, expressed as \( U_i = mgh \), where \( U_i \) is the initial potential energy, \( m \) is the mass, \( g \) is the acceleration due to gravity, and \( h \) is the height.
As he swings down, potential energy decreases, while kinetic energy increases. Ultimately, at the nadir of the swing, nearly all potential energy has converted to kinetic energy.
At the highest point, all energy is potential, expressed as \( U_i = mgh \), where \( U_i \) is the initial potential energy, \( m \) is the mass, \( g \) is the acceleration due to gravity, and \( h \) is the height.
As he swings down, potential energy decreases, while kinetic energy increases. Ultimately, at the nadir of the swing, nearly all potential energy has converted to kinetic energy.
Kinetic and Potential Energy
These are two fundamental energy types central to Tarzan's swing. **Potential Energy** is stored energy. It's determined initially by Tarzan's height above the ground. **Kinetic Energy** increases as he swings, reaching its peak when he is at the lowest point in the arc.
The equation for kinetic energy is \( K = \frac{1}{2}mv^2 \), where \( m \) is mass and \( v \) is velocity. In Tarzan's swing, initial potential energy equals the kinetic energy at the lowest point: \( U_i = K_f \).
This transformation showcases the principle of conservation of energy: energy is neither created nor destroyed, only changed from one form to another.
The equation for kinetic energy is \( K = \frac{1}{2}mv^2 \), where \( m \) is mass and \( v \) is velocity. In Tarzan's swing, initial potential energy equals the kinetic energy at the lowest point: \( U_i = K_f \).
This transformation showcases the principle of conservation of energy: energy is neither created nor destroyed, only changed from one form to another.
Centripetal Force
While Tarzan swings, a **centripetal force** keeps him moving in a curve rather than in a straight line. This force is directed toward the center of the circle's path Tarzan roughly follows.
At the average point of the swing, this force is significant and combines with the force of gravity. The centripetal force required to keep Tarzan moving along the vine is calculated as \( F_c = \frac{mv^2}{L} \), where \( m \) is mass, \( v \) is velocity, and \( L \) is the vine鈥檚 length.
Understanding centripetal force helps explain why the tension in the vine changes during the swing.
At the average point of the swing, this force is significant and combines with the force of gravity. The centripetal force required to keep Tarzan moving along the vine is calculated as \( F_c = \frac{mv^2}{L} \), where \( m \) is mass, \( v \) is velocity, and \( L \) is the vine鈥檚 length.
Understanding centripetal force helps explain why the tension in the vine changes during the swing.
Tension in a Vine
The tension in the vine is crucial to determining whether it will break during Tarzan's swing.
As Tarzan swings through the bottom of his arc, the tension in the vine is composed of two components:
As Tarzan swings through the bottom of his arc, the tension in the vine is composed of two components:
- The gravitational force (his weight).
- The centripetal force needed to keep him moving in a circle.
Angle Calculation
If the vine breaks, it's usually because the angle causes excessive tension. Calculating this angle involves understanding how tension varies with position and speed.
The tension at breaking requires solving for \[ T = mg \cos(\theta) + \frac{mv^2}{L} \]Set \( T = 950 \text{ N} \), which is the vine鈥檚 maximum strength, to determine \( \theta \). Solving this equation gives the critical angle at which the vine fails. This often involves combining insights from trigonometry and physics.
The tension at breaking requires solving for \[ T = mg \cos(\theta) + \frac{mv^2}{L} \]Set \( T = 950 \text{ N} \), which is the vine鈥檚 maximum strength, to determine \( \theta \). Solving this equation gives the critical angle at which the vine fails. This often involves combining insights from trigonometry and physics.
Breaking Point Analysis
Breaking points are pivotal in assessing safety limits. The vine breaks when the tension exceeds its capacity, which, in this case, is \(950 \, \text{N}\).
Analyzing the moment of breaking requires comprehending how tension accumulates based on Tarzan's speed and the vertical angle of the vine. You evaluate maximum stress during the swing to decide if and when the vine gives way. This involves identifying when the tension, influenced by both weight and centripetal forces, surpasses the vine's tensile strength.
Understanding breaking points is a reminder of how precise physics can guide safety in dynamic scenarios.
Analyzing the moment of breaking requires comprehending how tension accumulates based on Tarzan's speed and the vertical angle of the vine. You evaluate maximum stress during the swing to decide if and when the vine gives way. This involves identifying when the tension, influenced by both weight and centripetal forces, surpasses the vine's tensile strength.
Understanding breaking points is a reminder of how precise physics can guide safety in dynamic scenarios.
