A 1000 kg car has four 20 kg wheels. When the car is moving, what fraction of its total kinetic energy is due to rotation of the wheels about their axles? Assume that the wheels have the same rotational inertia as uniform disks of the same mass and size.
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Why do you not need to know the radius of the wheels?
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To determine the fraction of the car's total kinetic energy due to rotation of the wheels, we don't actually need to know the radius of the wheels. This is because the fraction of kinetic energy due to rotation depends only on the rotational inertia (or moment of inertia) of the rotating object, not on its radius.
Rotational inertia is a measure of how mass is distributed around an axis. For a uniform disk, the rotational inertia is given by the formula:
I = (1/2) * m * r^2
Where:
I is the rotational inertia
m is the mass of the object (wheel)
r is the radius of the object (wheel)
Since the question states that the wheels have the same rotational inertia as uniform disks of the same mass and size, we can assume that the rotational inertia is equal for all four wheels. Therefore, the fraction of kinetic energy due to rotation for each wheel is the same.
The total kinetic energy of an object can be calculated by summing up the rotational kinetic energy and translational kinetic energy. However, for the purposes of this question, we only need to consider the fraction due to rotation.
Since the rotational inertia is the same for all four wheels, the fraction of total kinetic energy due to rotation would be:
1/4 (one quarter) of the total kinetic energy.
Therefore, we can conclude that regardless of the radius of the wheels, the fraction of the car's total kinetic energy due to the rotation of the wheels about their axles is 1/4.