Radians Math Example 4

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Example 4

hard
A wheel of radius 3030 cm rotates at 120120 rpm (revolutions per minute). Find the angular velocity in radians per second and the linear speed of a point on the rim.

Solution

  1. 1
    Angular velocity: 120120 rpm =120×2π rad1 rev×1 min60 s=4π= 120 \times \frac{2\pi \text{ rad}}{1 \text{ rev}} \times \frac{1 \text{ min}}{60 \text{ s}} = 4\pi rad/s.
  2. 2
    Linear speed: v=rω=30×4π=120π≈376.99v = r\omega = 30 \times 4\pi = 120\pi \approx 376.99 cm/s.

Answer

ω=4π rad/s,v=120π≈377 cm/s\omega = 4\pi \text{ rad/s}, \quad v = 120\pi \approx 377 \text{ cm/s}
Angular velocity ω\omega in radians per second connects rotational motion to linear motion via v=rωv = r\omega. This clean relationship is another reason radians are the natural unit for angles in physics and mathematics. Converting rpm to rad/s requires multiplying by 2π60\frac{2\pi}{60}.

About Radians

A radian is an angle measurement defined by the arc length it subtends on a unit circle: one radian is the angle at which the arc length equals the radius. A full circle is 2π2\pi radians (about 6.28 radians), making radians the natural unit for trigonometry and calculus.

Learn more about Radians →

More Radians Examples

Example 1 easy

Convert [formula] to radians.

Example 2 medium

Find the arc length of a sector with radius [formula] cm and central angle [formula] radians.

Example 3 medium

Convert [formula] radians to degrees and identify which quadrant this angle is in.

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