Problem 58 - College Physics

Question

A \(55 \mathrm{~kg}\) woman is riding a 24 kg bike at a speed of \(7 \mathrm{~m} / \mathrm{s}\). The wheels of the bike can be treated as thin rings, each of mass \(3 \mathrm{~kg}\) and radius \(0.33 \mathrm{~m}\). What percentage of the total kinetic energy of the woman-bike system is carried in the rotational kinetic energy of the wheels?

Step-by-Step Solution

Answer

7.04% of the total kinetic energy is rotational.

Step 1: Calculate Total Mass

First, identify the total mass of the woman and the bike. The woman weighs 55 kg, and the bike weighs 24 kg without wheels. Since each wheel weighs 3 kg and there are 2 wheels, the bike's total mass is 24 kg. Thus, the total system mass is \(55 + 24 = 79\) kg.

Step 2: Calculate Translational Kinetic Energy

The translational kinetic energy (TKE) of a moving object is given by the formula \( KE_{ ext{trans}} = \frac{1}{2}mv^2 \), where \( m \) is the mass and \( v \) is the velocity. Here, \( m = 79 \) kg and \( v = 7 \) m/s. Thus, the translational kinetic energy \( KE_{ ext{trans}} = \frac{1}{2} \times 79 \times 7^2 = 1932.5 \text{ J} \).

Step 3: Determine Moment of Inertia for Each Wheel

The moment of inertia \( I \) for a thin ring is \( I = mr^2 \). Here, \( m = 3 \) kg for each wheel and \( r = 0.33 \) m. Thus, \( I = 3 \times (0.33)^2 = 0.3267 \text{ kg} \cdot \text{m}^2 \).

Step 4: Calculate Rotational Kinetic Energy of One Wheel

Rotational kinetic energy is given by \( KE_{ ext{rot}} = \frac{1}{2}I \omega^2 \). The angular velocity \( \omega \) is related to the linear velocity by \( \omega = \frac{v}{r} \). So, \( \omega = \frac{7}{0.33} = 21.21 \text{ rad/s} \). The rotational kinetic energy for one wheel is \( KE_{ ext{rot}} = \frac{1}{2} \times 0.3267 \times (21.21)^2 = 73.16 \text{ J} \).

Step 5: Calculate Total Rotational Kinetic Energy of Both Wheels

Since there are two wheels, the total rotational kinetic energy is \( 2 \times 73.16 = 146.32 \text{ J} \).

Step 6: Calculate Total Kinetic Energy of the System

The total kinetic energy (KE) is the sum of translational and rotational kinetic energies. Thus, \( KE_{ ext{total}} = 1932.5 + 146.32 = 2078.82 \text{ J} \).

Step 7: Find the Percentage of Rotational Kinetic Energy

To find the percentage of kinetic energy due to the rotational energy of the wheels, use the formula \( \text{Percentage} = \left( \frac{KE_{ ext{rot}}}{KE_{ ext{total}}} \right) \times 100 \). Substitute the values: \( \text{Percentage} = \left( \frac{146.32}{2078.82} \right) \times 100 = 7.04\% \).

Key Concepts

Translational Kinetic Energy

Translational kinetic energy is the energy an object possesses due to its motion along a path. It represents how much energy is in the object because it is moving in a straight line. This energy can be calculated using the formula:

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

where \( m \) is the mass of the object and \( v \) is its velocity.
In the context of the problem, the total mass of the system (woman and bike) is 79 kg, and it moves at 7 m/s.
This gives us a total translational kinetic energy of 1932.5 J. This energy depends on both how massive the object is and how fast it is moving.
When we think about problems involving kinetic energy, it helps to remember that as either mass or speed increases, the kinetic energy will increase.

Rotational Kinetic Energy

Rotational kinetic energy is similar to translational kinetic energy but occurs when an object rotates around an axis. When we think about motion, not all objects just slide around; some spin.
To calculate this type of energy, we have to consider how hard it is to rotate the object (its moment of inertia) and how fast it's spinning (angular velocity).
The formula for rotational kinetic energy is:

• Rotational Kinetic Energy (RKE) = \( \frac{1}{2}I \omega^2 \)

where \( I \) is the moment of inertia and \( \omega \) is the angular velocity.
In our problem, the wheels of the bike are spinning. Each wheel has a rotational kinetic energy of 73.16 J, and since both are spinning, the total rotational kinetic energy becomes 146.32 J.
This shows us that even though they're small compared to the whole system's motion, the wheels still contribute significantly to the overall energy.

Moment of Inertia

Moment of inertia is a bit like mass for rotational movements.
It tells us how difficult it is to change the rotation of an object—how much resistance the object offers to being spun. It depends both on the mass of the object and how that mass is distributed relative to the axis of rotation.
For example, for a thin ring or hoop, the moment of inertia is given by:

• Moment of Inertia (I) = \( mr^2 \)

where \( m \) is the mass and \( r \) is the radius from the axis of rotation.
Each wheel in the exercise has a moment of inertia of 0.3267 kg·m², calculated using its mass (3 kg) and radius (0.33 m). By understanding the moment of inertia, we grasp why some objects are easier to spin than others—even if they have the same mass.

Angular Velocity

Angular velocity describes how fast something is rotating around a point or axis. It's not just about how fast something is going, but how fast it's spinning.
Speed in a straight line is straightforward, but when it comes to spinning, we need to think about angles and how quickly those angles change.
It's represented by the symbol \( \omega \) and is calculated as:

• Angular Velocity (\( \omega \)) = \( \frac{v}{r} \)

where \( v \) is the linear velocity and \( r \) is the radius of the circle it's rotating around.
In the case of the bike wheels, with a speed of 7 m/s and a radius of 0.33 m, the angular velocity is 21.21 rad/s.
Understanding angular velocity helps us see how fast different parts of a rotating system move relative to each other and can be crucial in designing mechanical systems.