Question

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

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

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.

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.

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.

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.

Key concepts

Conservation of Energy

In physics, the conservation of energy is a crucial principle that assists in solving various problems. It asserts that energy in an isolated system cannot be created or destroyed; it merely changes form. In Tarzan's swinging scenario, this principle is pivotal.

To fully grasp how energy conservation applies here, note that Tarzan starts from a height with potential energy. This energy is calculated using the formula:

• Potential Energy (PE) = mgh

Where:

• \(m\) is mass
• \(g\) is the gravitational acceleration,
• \(h\) is the height.

As Tarzan swings downward, his potential energy converts into kinetic energy, which expresses his velocity at the lowest point. This is described as:

• Kinetic Energy (KE) = \(\frac{1}{2} mv^2\)

Applying the conservation of energy for Tarzan's swing, set his initial potential energy equal to his kinetic energy at the bottom, leading to the equation used in the step-by-step solution:

• \(v = \sqrt{2gh}\)

This equation helps determine Tarzan's speed as he swings to the bottom, crucial for further calculations.

Tension Calculation

Calculating tension is vital to decide if Tarzan's vine will endure the swing. Tension refers to the force exerted by a string, rope, or vine in this case. The key to calculating tension in Tarzan's swing is understanding how it changes with motion.

The formula for tension in this situation is:

• \(T = mg + \frac{mv^2}{r}\)

Where:

• \(m\) represents the mass of Tarzan extracted from his weight.
• \(g\) is the acceleration due to gravity.
• \(v\) is the velocity calculated using the conservation of energy.
• \(r\) is the radius of the swing, in this example equal to the length of the vine.

This formula comprises two parts:

• The gravitational force \(mg\) exerted on Tarzan.
• The centripetal force \(\frac{mv^2}{r}\) necessary to keep Tarzan moving in a circular path during his swing.

By finding the total tension, we can compare it against the vine's breaking strength to assess if it's safe.

Kinetic and Potential Energy

Kinetic and potential energies are fundamental concepts in mechanics. They help explain how energy swaps forms during Tarzan’s swing.

**Potential Energy**

• This is the stored energy based on position. At the top of Tarzan's swing, he has maximum potential energy. Calculated as \(PE = mgh\), it is dependent on his height and mass.

**Kinetic Energy**

• When Tarzan starts to swing, his potential energy transforms into kinetic energy, the energy of motion, quantified as \(KE = \frac{1}{2} mv^2\). At the swing's nadir, he reaches maximal kinetic energy while having minimal potential energy.

Understanding the interplay between kinetic and potential energy is key for analyzing motion, enabling us to determine Tarzan’s velocity at different points during his swing.

Gravitational Force

Gravitational force is the attractive pull exerted by the Earth on objects. It's a core component of many physics problems involving motion, including Tarzan's swing.

When Tarzan is at the top of the cliff, gravity pulls him downward, converting his potential energy into kinetic energy during the swing. The gravitational force is expressed by the formula:

• \(F = mg\)

Where:

• \(m\) is Tarzan's mass
• \(g\) is the gravitational acceleration, typically \(32 \mathrm{ft/s^2}\) on Earth.

Gravitational force determines how fast Tarzan accelerates as he swings, impacting both his speed and the tension in the vine. It is integral in the equations used for calculations, guiding the assessment of whether the vine can withstand the forces during Tarzan's adventurous swing.