## Rotational Variables

10.1 Rotational Variables

### Learning Objectives

By the end of this section, you will be able to:

- Describe the physical meaning of rotational variables as applied to fixed-axis rotation
- Explain how angular velocity is related to tangential speed
- Calculate the instantaneous angular velocity given the angular position function
- Find the angular velocity and angular acceleration in a rotating system
- Calculate the average angular acceleration when the angular velocity is changing
- Calculate the instantaneous angular acceleration given the angular velocity function

### Angular Position

In the figure above, the red dot is a particle that moves in uniform circular motion. Its position vector from the origin of the circle to the particle sweeps out the angle 𝜃, which increases in the counterclockwise direction as the particle moves along its circular path. The angle 𝜃 is called the **angular position** of the particle.

The angle is related to the radius of the circle and the arc length by:

The angular displacement is the change in angular position: .

### Angular Velocity

The magnitude of the **angular velocity**, denoted by 𝜔 (omega), is the time rate of change of the angle 𝜃 as the particle moves in its circular path.

The **instantaneous angular velocity** is defined as the limit in which in the average angular velocity:

### Angular Acceleration

We need to define an **angular acceleration** for describing situations where 𝜔 changes. The faster the change in 𝜔, the greater the angular acceleration.

We define the **instantaneous angular acceleration** 𝛼 as the derivative of angular velocity with respect to time:

**Practice!**

Practice 10.1.1 |
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A disk that initially spins at 2 revolutions/sec is braked uniformly to a stop in half a second. What is its initial angular speed ω _{0}? |

(a) π/2 rad/s |

(b) π rad/s |

(c) 2π rad/s |

(d) 4π rad/s |

(e) 8π rad/s |

Practice 10.1.2 |
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A disk that initially spins at 2 revolutions/sec is braked uniformly to a stop in half a second. What is the angular acceleration α? |

(a) -4π rad/s^{2} |

(b) 4π rad/s^{2} |

(c) -8π rad/s^{2} |

(d) 8π rad/s^{2} |

(e) -2π rad/s^{2} |

Practice 10.1.3 |
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The graph shows the angular velocity ω and angular acceleration α versus time t for a rotating body. At which of the following times is the rotation speeding up at the greatest rate? |

(a) t = 1 s |

(b) t = 2 s |

(c) t = 3 s |

(d) t = 4 s |

(e) t = 5 s |

### Relating Linear and Angular Measurements

Earlier in this module we defined the angular position as

Now let’s rearrange this to solve for the arc length, s:

The arc length, s, is measured in units of meters and is a **linear** measurement, while 𝜃 is measured in units of rad (radians) and is an **angular** measurement. This expression relates the linear position to the angular position.

Similarly, the linear speed (*v*) of a particle is related to the angular speed by: and the linear acceleration (*a*) is related to the angular acceleration by: .

Pause & Predict 10.1.1 |
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Two wheels are connected by a belt. Wheel 1 rotates counterclockwise at 6.4 rad/s. What is the angular speed of wheel 2? |

(a) 6.4 rad/s |

(b) 4.3 rad/s |

(c) 21.8 rad/s |

(d) 14.7 rad/s |

(e) 3.2 rad/s |

Pause & Predict 10.1.2 |
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The pepper is 0.25 m from the center of the turntable, and the salt is 0.20 m from the center. If the turntable has an angular acceleration of 0.6 rad/s^{2} (ccw), what is the linear acceleration of the salt? |

(a) 0.12 m/s^{2} |

(b) 0.15 m/s^{2} |

(c) 3.0 m/s^{2} |

(d) 2.4 m/s^{2} |

(e) 0.60 m/s^{2} |

**Practice!**

Practice 10.1.4 |
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Bonnie sits on the outer rim of a merry-go-round, and Clyde sits midway between the center and the rim. The merry-go-round makes one complete revolution every two seconds. Clyde’s angular velocity is: |

(a) the same as Bonnie’s |

(b) twice Bonnie’s |

(c) half Bonnie’s |

Practice 10.1.5 |
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Compared to a gear tooth on the rear sprocket (on the left, of small radius) of a bicycle, a gear tooth on the front sprocket (on the right, of large radius) has |

(a) a faster linear speed and a faster angular speed. |

(b) the same linear speed and a faster angular speed. |

(c) a slower linear speed and the same angular speed. |

(d) the same linear speed and a slower angular speed. |

(e) none of the above. |

**Discuss!**

Rebecca has gone to a conference in Albany, NY, leaving Brent at home in Dahlonega to look after Max the dog.

(a) Which one them has the greater angular speed, ω?

(b) Which of them has the greater tangential speed, v ?

The small wheel in the picture is rotating at 50 rpm. What is the angular speed of the large wheel if its radius is twice that of the small wheel.