3 questions about Angular velocity equation

1. An object moving in a circular path has a radius of 0.5 m. The particle is able to cover an angle of 60π rad in 15 seconds. Determine the angular speed of the object.

__Known:__

Radius (r) = 0.5 meters

Angle (θ) = 60π rad

Time interval (t) = 15 seconds

__Wanted:__ Angular velocity (ω)

__Solution:__

θ = ω t

60π = ω (15)

ω = 60π / 15 = 4π rad/s

2. A bicycle wheel with a radius of 30 cm rotates at a constant speed. The tire valve can rotate at an angle of 120π rad for 10 seconds. Determine the angular speed.

__Known:__

Radius (r) = 30 cm

Angle (θ) = 120π rad

Time interval (t) = 10 s

__Wanted:__ Angular velocity (ω)

__Solution:__

θ = ω t

120π = ω (10)

ω = 120π / 10 = 12π rad/s

3. The rope is wrapped around a wheel with radius R = 25 cm as shown. If a point on the rope (point A) has a speed of 5 m/s then the rotational speed of the wheel is…

__Known:__

Radius (r) = 25 cm = 0.25 meters

Speed (v) = 5 m/s

__Wanted:__ Angular velocity (ω)

__Solution:__

The formula for the relation between angular speed, linear speed and radius:

v = r ω

5 = (0,25) ω

ω = 5 / 0,25 = 20 rad/s