1. A ball at the end of a string is revolving uniformly in a horizontal circle of radius 2 meters at constant angular speed 10 rad/s. Determine the magnitude of the linear velocity of a point located :
(a) 0.5 meters from the center
(b) 1 meter from the center
(c) 2 meters from the center
Known :
Radius (r) = 0.5 meters, 1 meter, 3 meters
The angular speed = 10 radians/second
Wanted : The linear velocity
Solution :
v = r ω
v = the linear velocity, r = radius, ω = the angular velocity
(a) The linear velocity (v) of a point located at r = 0.5 meters
v = r ω = (0.5 meters)(10 rad/s) = 5 meters/second
(b) The linear velocity (v) of a point located at r = 1 meter
v = r ω = (1 meter)(10 rad/s) = 10 meters/second
(c) The linear velocity (v) of a point located at r = 2 meters
v = r ω = (2 meters)(10 rad/s) = 20 meters/second
2. The blades in a blender rotate at a rate of 5000 rpm. Determine the magnitude of the linear velocity :
(a) a point located 5 cm from the center
(b) a point located 10 cm from the center
Known :
Radius (r) = 5 cm and 10 cm
The angular speed (ω) = 5000 revolutions / 60 seconds = 83.3 revolutions / second = (83.3)(6.28 radian) / second = 523.3 radians / second
Wanted : The magnitude of the linear velocity
Solution :
(a) The magnitude of the linear velocity of a point located 0.05 m from the center
v = r ω = (0.05 m)(523.3 rad/s) = 26 m/s
(b) The magnitude of the linear velocity of a point located 0,1 m from the center
v = r ω = (0.1 m)(523.3 rad/s) = 52 m/s
3. A point on the edge of a wheel 30 cm in radius, around a circle at constant speed 10 meters/second.
What is the magnitude of the angular velocity?
Known :
Radius (r) = 30 cm = 0.3 meters
The linear velocity (v) = 10 meters/second
Wanted : the angular velocity
Solution :
ω = v / r = 10 / 0.3 = 33 radians/second
4. A car with tires 50 cm in diameter travels 10 meters in 1 second. What is the angular speed ?
Known :
Radius (r) = 0.25 meter
The linear speed of a point on the edge of tires (v) = 10 meters/second
Wanted: The angular speed
Solution :
ω = v / r = 10 / 0.25 = 40 radians/second
5. The angular speed of wheel 20 cm in radians is 120 rpm. What is the distance if the car travels in 10 seconds.
Known :
Radius (r) = 20 cm = 0.2 meters
The angular speed = 120 rev / 60 seconds = 2 rev / second = (2)(6.28) radians / second = 12.56 radians / second
Wanted : distance
Solution :
Velocity of the edge of wheel :
v = r ω = (0.2 meters)(12.56 radians/second) = 2.5 meters/second
2.5 meters / second means a point on the edge of wheel travels 2.5 meters each 1 second. After 10 seconds, the point travels 25 meters.
So the distance is 25 meters.
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