Answer:
Angular velocity: [tex]\omega = \frac{7\pi}{18} \,\frac{rad}{s}[/tex] ([tex]1.222\,\frac{rad}{s}[/tex])
Linear velocity: [tex]v = \frac{14\pi}{3}\,\frac{m}{s}[/tex] ([tex]14.661\,\frac{m}{s}[/tex])
Step-by-step explanation:
The gear experiments a pure rotation with axis passing through its center, the angular ([tex]\omega[/tex]), in radians per second, and linear velocities ([tex]v[/tex]), in inches per second, of a point on the outer edge of the element are, respectively:
[tex]\omega = \frac{2\pi}{60}\cdot \dot n[/tex] (1)
[tex]v = R\cdot \omega[/tex] (2)
Where:
[tex]\dot n[/tex] - Rotation rate, in revolutions per minute.
[tex]R[/tex] - Radius of the gear, in inches.
If we know that [tex]\dot n = \frac{35}{3}\,\frac{rev}{min}[/tex] and [tex]R = 12\,in[/tex], then the linear and angular velocities of the gear are, respectively:
[tex]\omega = \frac{2\pi}{60}\cdot \dot n[/tex]
[tex]\omega = \frac{7\pi}{18} \,\frac{rad}{s}[/tex] ([tex]1.222\,\frac{rad}{s}[/tex])
[tex]v = R\cdot \omega[/tex]
[tex]v = \frac{14\pi}{3}\,\frac{m}{s}[/tex] ([tex]14.661\,\frac{m}{s}[/tex])
