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Ch. 10 - Rotational Motion
Giancoli Douglas - Physics for Scientists and Engineers 5th edition
Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoGiancoli Douglas 5th editionCh. 10 - Rotational MotionProblema 79b
Capitolo 10, Problema 79b

The 1100-kg mass of a car includes four tires, each of mass 35 kg (including wheels) and diameter 0.80 m. Assume each tire and wheel combination acts as a solid cylinder. Determine the fraction of the kinetic energy in the tires and wheels.

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Determine the total mass of the car excluding the tires. Subtract the combined mass of the four tires from the total mass of the car: \( m_{car} = 1100 \ \text{kg} - 4 \times 35 \ \text{kg} \).
Calculate the moment of inertia of a single tire, treating it as a solid cylinder. The formula for the moment of inertia of a solid cylinder rotating about its central axis is \( I = \frac{1}{2} m r^2 \), where \( m \) is the mass of the tire and \( r \) is its radius. Use the given diameter of 0.80 m to find the radius \( r = \frac{0.80}{2} \ \text{m} \).
Determine the rotational kinetic energy of all four tires. The rotational kinetic energy of a single tire is given by \( KE_{rot} = \frac{1}{2} I \omega^2 \), where \( \omega \) is the angular velocity. Since all four tires are identical, multiply the rotational kinetic energy of one tire by 4.
Calculate the translational kinetic energy of the car. The formula for translational kinetic energy is \( KE_{trans} = \frac{1}{2} m v^2 \), where \( m \) is the total mass of the car (including the tires) and \( v \) is the linear velocity. Note that the linear velocity \( v \) is related to the angular velocity \( \omega \) of the tires by \( v = r \omega \).
Find the fraction of the total kinetic energy that is in the tires and wheels. The total kinetic energy is the sum of the translational kinetic energy and the rotational kinetic energy of the tires. The fraction is given by \( \text{Fraction} = \frac{KE_{rot}}{KE_{total}} \), where \( KE_{total} = KE_{trans} + KE_{rot} \).

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Kinetic Energy

Kinetic energy is the energy an object possesses due to its motion, calculated using the formula KE = 1/2 mv², where m is the mass and v is the velocity of the object. In the context of the car, both the translational kinetic energy of the car's mass and the rotational kinetic energy of the tires must be considered to determine the total kinetic energy.
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Intro to Rotational Kinetic Energy

Moment of Inertia

Moment of inertia is a measure of an object's resistance to changes in its rotation and depends on the mass distribution relative to the axis of rotation. For solid cylinders, the moment of inertia can be calculated using the formula I = 1/2 m r², where m is the mass and r is the radius. This concept is crucial for understanding how the tires contribute to the overall kinetic energy of the car.
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Intro to Moment of Inertia

Fraction of Kinetic Energy

The fraction of kinetic energy refers to the proportion of the total kinetic energy that is attributed to a specific part of a system, in this case, the tires and wheels of the car. To find this fraction, one must calculate the kinetic energy of the tires and wheels separately and then divide it by the total kinetic energy of the car, allowing for a clear understanding of how much energy is involved in the motion of the tires.
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Kinetic Friction Problems
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