Problem 90 - University Physics with Modern Physics

Question

Tarzan and Jane in the 21 st Century. Tarzan has foolishly gotten himself into another scrape with the animals and must be rescued once again by Jane. The 60.0 -kg Jane starts from rest at a height of 5.00 \(\mathrm{m}\) in the trees and swings down to the ground using a thin, but very rigid, 30.0 -kg vine 8.00 \(\mathrm{m}\) long. She arrives just in time to snatch the 72.0 -kg Tarzan from the jaws of an angry hippopotamus. What is Jane's (and the vine's) angular speed (a) just before she grabs Tarzan and (b) just after she grabs him? (c) How high will Tarzan and Jane go on their first swing after this daring rescue?

Step-by-Step Solution

Answer

(a) 0.725 rad/s, (b) 0.402 rad/s, (c) 0.524 m.

Step 1: Calculate Potential Energy at Top

Jane begins at a height of 5.00 m. Calculate her gravitational potential energy using \( PE = mgh \), where \( m = 60.0 \; \text{kg} \), \( g = 9.81 \; \text{m/s}^2 \), and \( h = 5.00 \; \text{m} \). The potential energy at this height is \( PE = 60.0 \times 9.81 \times 5.00 = 2943 \; \text{J} \).

Step 2: Convert Potential Energy to Kinetic Energy

As Jane swings down, her potential energy is converted into kinetic energy. Just before she grabs Tarzan, the kinetic energy \( KE \) is equal to the initial potential energy \( PE \), which is 2943 J.

Step 3: Calculate Angular Speed Before Grabbing Tarzan

Using the formula for kinetic energy \( KE = \frac{1}{2} I \omega^2 \), where \( I \) is the moment of inertia of Jane and the vine combined. \( I = m r^2 \), \( m = 60.0 + 30.0 \; \text{kg} \) (Jane plus vine) and \( r = 8.00 \; \text{m} \). Calculate \( I = 90.0 \times 8.00^2 = 5760 \; \text{kg} \cdot \text{m}^2 \). Solve for \( \omega \): \( 2943 = \frac{1}{2} \times 5760 \times \omega^2 \), hence \( \omega = \sqrt{\frac{2 \times 2943}{5760}} = 0.725 \; \text{rad/s} \).

Step 4: Calculate Angular Speed After Grabbing Tarzan

Now consider the total mass after Jane grabs Tarzan, which is \( m = 60.0 + 30.0 + 72.0 = 162.0 \; \text{kg} \). The new moment of inertia is \( I = 162.0 \times 8.00^2 = 10368 \; \text{kg} \cdot \text{m}^2 \). Use conservation of angular momentum: \( I_i \omega_i = I_f \omega_f \), where \( I_i = 5760 \; \text{kg} \cdot \text{m}^2 \), \( \omega_i = 0.725 \; \text{rad/s} \), and \( I_f = 10368 \; \text{kg} \cdot \text{m}^2 \). Solve for \( \omega_f \): \( \omega_f = \frac{I_i \omega_i}{I_f} = \frac{5760 \times 0.725}{10368} = 0.402 \; \text{rad/s} \).

Step 5: Calculate Maximum Height After Swing

After Jane grabs Tarzan, the total mechanical energy is conserved. Use \( KE = \frac{1}{2} I_f \omega_f^2 \), where \( \omega_f = 0.402 \; \text{rad/s} \) and \( I_f = 10368 \; \text{kg} \cdot \text{m}^2 \). The kinetic energy is \( KE = \frac{1}{2} \times 10368 \times 0.402^2 = 836.92 \; \text{J} \). Convert this \( KE \) back to potential energy at the highest point using \( PE = mgh \), \( m = 162.0 \; \text{kg} \), and solve for \( h \): \( 836.92 = 162.0 \times 9.81 \times h \), hence \( h = \frac{836.92}{162.0 \times 9.81} = 0.524 \; \text{m} \).

Key Concepts

Gravitational Potential Energy

Gravitational potential energy is a form of energy that is stored due to an object's position in a gravitational field. It depends on three main factors: the mass of the object, the height of the object above a reference point, and the gravitational constant (usually denoted as \( g = 9.81 \, \text{m/s}^2 \) on Earth). The formula for calculating gravitational potential energy (PE) is \( PE = mgh \), where \( m \) is the mass, \( g \) is the acceleration due to gravity, and \( h \) is the height.

In the context of the problem with Jane, she starts at a height of \( 5.00 \, \text{m} \) in the trees. By using the formula, her potential energy is calculated as \( 2943 \, \text{J} \) when her mass is \( 60.0 \, \text{kg} \). This potential energy represents the energy Jane has due to her elevated position before she swings down to the ground.

Kinetic Energy

Kinetic energy is the energy of motion. When an object is moving, it possesses kinetic energy. The amount of kinetic energy depends on the mass of the object and its velocity. It can be calculated using the formula \( KE = \frac{1}{2} mv^2 \), where \( m \) is the mass and \( v \) is the velocity of the object.

For Jane, as she swings down from the tree, her gravitational potential energy gets converted into kinetic energy. Just before she reaches the ground and grabs Tarzan, her kinetic energy is equal to the potential energy she initially had at the top of the swing (\( 2943 \, \text{J} \)). This conversion is a principle of conservation of energy, meaning energy changes form but the total amount remains constant during her swing.

Angular Momentum Conservation

Angular momentum conservation is a fundamental concept in physics that states the total angular momentum of a closed system remains constant if no external torque acts on it. Angular momentum is the rotational analogue of linear momentum and is given by \( L = I \omega \), where \( I \) is the moment of inertia and \( \omega \) is the angular velocity.

In the problem with Tarzan and Jane, once Jane grabs Tarzan, the system's mass changes, which affects the moment of inertia. However, angular momentum conservation means the product of the initial moment of inertia \( I_i \) and initial angular speed \( \omega_i \) must equal the product of the final moment of inertia \( I_f \) and final angular speed \( \omega_f \). The initial setup before grabbing Tarzan and the final configuration after grabbing him both adhere to this conservation principle.

Moment of Inertia

The moment of inertia is an important quantity in rotational dynamics. It describes how an object's mass is distributed in relation to the axis about which it rotates. Essentially, it indicates how difficult it is to change the rotational state of an object, akin to mass in linear motion. The moment of inertia depends on the mass of the object and the square of the distance from the rotation axis (\( I = mr^2 \) for simple systems).

For the scenario described, Jane and the vine initially have a moment of inertia calculated with a total mass of \( 90.0 \, \text{kg} \) and the vine's length of \( 8.00 \, \text{m} \). This gives us an initial \( I \) of \( 5760 \, \text{kg} \cdot \text{m}^2 \). After grabbing Tarzan, the combined mass increases, raising the moment of inertia to \( 10368 \, \text{kg} \cdot \text{m}^2 \). This increase directly affects the angular velocity as shown in the problem using the principle of angular momentum conservation.