A wheel rotates at a frequency of 6 revolutions per second. The angular velocity of this wheel can be expressed...

1. TRANSLATE the problem information

• Given information:
  • Wheel frequency = \(\mathrm{6\ revolutions\ per\ second}\)
  • Angular velocity can be expressed as \(\mathrm{b\pi\ radians\ per\ second}\)
  • Need to find the value of constant b

2. INFER the conversion approach

• Key insight: We need to convert from frequency (revolutions/second) to angular velocity (radians/second)
• This requires the fundamental relationship: \(\mathrm{1\ revolution = 2\pi\ radians}\)
• Therefore: \(\mathrm{angular\ velocity = 2\pi \times frequency}\)

3. Apply the conversion formula

• Angular velocity = \(\mathrm{2\pi \times 6\ revolutions\ per\ second}\)
• Angular velocity = \(\mathrm{12\pi\ radians\ per\ second}\)

4. SIMPLIFY to find b

• We know: angular velocity = \(\mathrm{b\pi\ radians\ per\ second}\)
• We calculated: angular velocity = \(\mathrm{12\pi\ radians\ per\ second}\)
• Setting them equal: \(\mathrm{b\pi = 12\pi}\)
• Therefore: \(\mathrm{b = 12}\)

Answer: B

Why Students Usually Falter on This Problem

Most Common Error Path:

Missing conceptual knowledge: Students don't remember that \(\mathrm{1\ revolution = 2\pi\ radians}\)

Without this conversion factor, they might think angular velocity equals frequency directly, leading them to conclude \(\mathrm{b = 6}\). This may lead them to select Choice A (6).

Second Most Common Error:

Weak TRANSLATE skill: Students misread the problem and think they can use frequency directly as angular velocity

They see "\(\mathrm{6\ revolutions\ per\ second}\)" and "\(\mathrm{b\pi\ radians\ per\ second}\)" and incorrectly assume these should be equal without conversion, again leading to \(\mathrm{b = 6}\). This may lead them to select Choice A (6).

The Bottom Line:

This problem tests whether students understand the fundamental relationship between rotational motion units. The key insight is recognizing that revolutions must be converted to radians using the \(\mathrm{2\pi}\) factor before comparing with the given angular velocity expression.