Angular velocity and linear velocity – problems and solutions

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

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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

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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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