Problem 81 - University Physics with Modern Physics

Question

If a person of mass \(M\) simply moved forward with speed \(V\), his kinetic energy would be \\( \frac{1}{2} \\)\(MV^2\). However, in addition to possessing a forward motion, various parts of his body (such as the arms and legs) undergo rotation. Therefore, his total kinetic energy is the sum of the energy from his forward motion plus the rotational kinetic energy of his arms and legs. The purpose of this problem is to see how much this rotational motion contributes to the person's kinetic energy. Biomedical measurements show that the arms and hands together typically make up 13% of a person's mass, while the legs and feet together account for 37%. For a rough (but reasonable) calculation, we can model the arms and legs as thin uniform bars pivoting about the shoulder and hip, respectively. In a brisk walk, the arms and legs each move through an angle of about \(\pm30^\circ\) (a total of 60\(^\circ\)) from the vertical in approximately 1 second. Assume that they are held straight, rather than being bent, which is not quite true. Consider a 75-kg person walking at 5.0 km/h, having arms 70 cm long and legs 90 cm long. (a) What is the average angular velocity of his arms and legs? (b) Using the average angular velocity from part (a), calculate the amount of rotational kinetic energy in this person's arms and legs as he walks. (c) What is the total kinetic energy due to both his forward motion and his rotation? (d) What percentage of his kinetic energy is due to the rotation of his legs and arms?

Step-by-Step Solution

Answer

(a) \( \omega = \frac{\pi}{3} \) rad/s. (b) Calculate \( KE_{rot} \) using given masses and lengths. (c) Add \( KE_{trans} \) and \( KE_{rot} \). (d) Calculate the percentage using \( \frac{KE_{rot}}{KE_{total}} \).

Step 1: Convert Speed to Meters per Second

The person's speed is given as 5.0 km/h. We need to convert this speed to meters per second for calculations involving kinetic energy.Use the conversion factor: 1 km/h = \( \frac{1000}{3600} \) m/s. Therefore, 5.0 km/h = 5.0 × \( \frac{1000}{3600} \) = \( \frac{50}{36} \) = 1.39 m/s approximately.

Step 2: Calculate Average Angular Velocity

The angular motion of the arms and legs is specified to cover an angle of 60° in one second. To find the angular velocity, we need to convert this angle from degrees to radians because angular velocity is typically measured in radians per second.60° = \( \frac{60 \pi}{180} \) = \( \frac{\pi}{3} \) radians.Thus, the average angular velocity \( \omega \) is \( \frac{\Delta \theta}{\Delta t} = \frac{\pi/3}{1} \) = \( \frac{\pi}{3} \) rad/s.

Step 3: Calculate Rotational Inertia

The arms and hands make up 13% of the person's mass, and the legs and feet 37%. Model the arms and legs as thin uniform bars.Formula for the moment of inertia of a thin rod about an end: \( I = \frac{1}{3}mL^2 \)Calculate mass of arms: \( M_{arms} = 0.13 \times 75 \) kg = 9.75 kg.Calculate mass of legs: \( M_{legs} = 0.37 \times 75 \) kg = 27.75 kg.Calculate inertia for arms: \( I_{arms} = \frac{1}{3} \times 9.75 \times (0.7)^2 \)Calculate inertia for legs: \( I_{legs} = \frac{1}{3} \times 27.75 \times (0.9)^2 \)

Step 4: Calculate Rotational Kinetic Energy

The formula for rotational kinetic energy is:\( KE_{rot} = \frac{1}{2} I \omega^2 \).Using the \( \omega \) from Step 2 and inertias from Step 3:- For arms: \( KE_{arms} = \frac{1}{2} I_{arms} \left(\frac{\pi}{3}\right)^2 \).- For legs: \( KE_{legs} = \frac{1}{2} I_{legs} \left(\frac{\pi}{3}\right)^2 \).Add both kinetic energies to find total rotational kinetic energy.

Step 5: Calculate Total Kinetic Energy

The total kinetic energy is the sum of translational and rotational kinetic energies.Translational kinetic energy \( KE_{trans} = \frac{1}{2} MV^2 \), with speed \( V = 1.39 \) m/s.\( KE_{trans} = \frac{1}{2} \times 75 \times (1.39)^2 \).Total kinetic energy is sum of \( KE_{trans} \) and \( KE_{rot} \).

Step 6: Calculate Percentage of Rotational Kinetic Energy

To find the percentage of total kinetic energy that is rotational:\( \frac{KE_{rot}}{KE_{total}} \times 100 \% \).This gives the fraction of the total kinetic energy that is due to rotation.

Key Concepts

Moment of Inertia

The moment of inertia is a fundamental concept in rotational mechanics, representing the rotational equivalent of mass in linear motion. It quantifies how difficult it is to change an object's rotational state. The larger the moment of inertia, the harder it is to make that object spin or stop spinning.

• The formula for the moment of inertia of a thin rod rotating about one of its ends is given by: \( I = \frac{1}{3}mL^2 \), where \( m \) is the mass, and \( L \) is the length of the rod.
• In the exercise, we treat arms and legs as thin uniform bars. We use this model because it simplifies the calculation and provides a reasonable approximation of how the mass is distributed.
• By substituting values specific to a person’s limbs (e.g., mass of arms and legs), we can calculate the moment of inertia tailored to each limb's specifics.

For a person with arms and legs modeled as described, this concept helps us understand how the limbs contribute to rotational motion, significantly impacting the overall kinetic energy during activities like walking.

Angular Velocity

Angular velocity is crucial when studying rotational motion, acting as the rotational counterpart to linear velocity. It measures how fast an object rotates or revolves relative to another point, and is usually expressed in radians per second (rad/s).

• In the context of the exercise, the arms and legs of a walking person swing back and forth, creating angular motion. This movement is described by angular velocity.
• To find the average angular velocity, we consider the range of motion: arms and legs move through a total angle of 60° (or \( \frac{\pi}{3} \) radians) in approximately one second, resulting in an angular velocity of \( \frac{\pi}{3} \text{ rad/s} \).
• Understanding angular velocity in human motion allows us to calculate the rotational kinetic energy of moving limbs, as it directly influences how much energy is involved in rotation.

Angular velocity provides insight into the dynamics of rotational movements in various activities, whether in athletics, everyday activities, or even mechanical systems.

Translational Kinetic Energy

Translational kinetic energy refers to the energy due to the motion of an object moving along a path, and it's an essential concept in physics. For linear motion, it's given by the formula: \( KE_{trans} = \frac{1}{2} MV^2 \), where \( M \) is the mass and \( V \) is the velocity of the object.

• In the presented exercise, we deal with a person's translational kinetic energy as they walk forward. This energy is distinct from rotational energy, which is associated with the limb movements.
• With a speed of 1.39 m/s (converted from 5 km/h), the translational kinetic energy of the individual provides a baseline measurement of how much energy is involved in their forward motion.
• This energy contributes substantially to the overall kinetic energy of the person, but understanding it alongside rotational energy gives a clearer picture of total energy involved in motion.

By examining both translational and rotational kinetic energies, we gain a comprehensive understanding of how different types of motion contribute to the total energy expenditure during activities like walking.