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Chapter 12: Problem 20

Tarzan, who weighs \(180 \mathrm{lb}\), swings from a cliff at the end of a convenient 50 -ft vine; see Fig. \(12-21 .\) From the top of the cliff to the bottom of the swing, Tarzan would fall by \(8.5 \mathrm{ft}\). The vine has a breaking strength of 250 lb. Will the vine break?

Short Answer

Expert verified
The short answer will be whether the vine breaks or not, depending on the comparison in step 4. The key point here is that the vine will not break if the maximum tension is less than or equal to the breaking strength of the vine. If the tension is greater, it means that the force to keep the swing in circular motion exceeds the vine's strength at the lowest point of the swing, which can result in the vine breaking.

Step by step solution

01

Identify Given Values

Identify the values given:- The weight of Tarzan is \(180 \mathrm{lb}\), the falling height is \(8.5 \mathrm{ft}\), and the strength of the vine is \(250 \mathrm{lb}\). The gravitational force is \(32 \mathrm{ft/s^2}\). All these values are needed for the calculations.
02

Find the Velocity at bottom of Swing

Tarzan's velocity at the bottom of the swing (v) can be found using the principle of conservation of energy. According to this principle, the energy at the top of the swing is equal to the energy at the bottom of the swing. In other words, Tarzan's kinetic energy at the bottom of the swing equals his potential energy at the top of swing. Therefore, \(v = \sqrt{2gh}\) , where \(g\) represents the gravity and \(h\) the falling height.
03

Calculate the Tension in the Vine

The maximum tension (T) in the vine can be calculated using the formula: \(T = mg + \frac{mv^2}{r}\). In this equation, \(m\) is the mass, \(r\) is the radius of the circle (equivalent to falling height here), \(v\) is the velocity, and \(g\) is the gravity acceleration. In this case, \(m\) should be calculated using Tarzan's weight divided by \(g\), and \(v\) is obtained from step 2. The output (T) is the maximum tension in the vine.
04

Compare Tension with Breaking Strength

Compare the maximum tension (T) obtained from Step 3 with the breaking strength of the vine. If T is greater than the strength of the vine, the vine will break. If not, the vine will not break.

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