## Uniform Circular Motion

4.4 Uniform Circular Motion

### Learning Objectives

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

- Solve for the centripetal acceleration of an object moving on a circular path.
- Use the equations of circular motion to find the position, velocity, and acceleration of a particle executing circular motion.
- Explain the differences between centripetal acceleration and tangential acceleration resulting from nonuniform circular motion.
- Evaluate centripetal and tangential acceleration in nonuniform circular motion, and find the total acceleration vector.

### Centripetal Acceleration

In two- and three-dimensional kinematics, even if the speed is a constant, a particle can have acceleration if it moves along a curved trajectory such as a circle. In this case the velocity vector is changing, or .

A particle moving in a circle at a constant speed has an acceleration with magnitude

This is a radial acceleration and is called the **centripetal acceleration**, which is why we give it the subscript *c*. The word *centripetal* comes from the Latin words *centrum* (meaning “center”) and *petere* (meaning “to seek”), and thus takes the meaning “center seeking.”

Pause and Predict 4.4.1 |
---|

What is the moon’s acceleration as it orbits Earth? |

(a) 1.4 × 10^{-7} m/s^{2} |

(b) 2.7 × 10^{-6} m/s^{2} |

(c) 2.7 × 10^{-3} m/s^{2} |

(d) 2.0 × 10^{7} m/s^{2} |

(e) 5.2 × 10^{5} m/s^{2} |

Pause and Predict 4.4.2 |
---|

If the tangential acceleration is 100 m/s^{2}, what is the total acceleration of the yo-yo at point 2? |

(a) 780 m/s^{2} |

(b) 790 m/s^{2} |

(c) 880 m/s^{2} |

(d) 680 m/s^{2} |

(e) 300 m/s^{2} |

**Practice!**

Practice 4.4.1 |
---|

Is it possible for an object to have a nonzero acceleration if the object is traveling at constant ?velocity |

(a) Yes |

(b) No |

(c) Need more information to answer |

Practice 4.4.2 |
---|

Is it possible for an object to have a nonzero acceleration if the object is traveling at constant ?speed |

(a) Yes |

(b) No |

(c) Need more information to answer |

Practice 4.4.3 |
---|

A particle moves along the circular path shown, with constant speed. Its velocity vector at two different times is shown. What is the direction of the acceleration when the particle is at point x? |

(a) Direction A |

(b) Direction B |

(c) Direction C |

(d) Direction D |

(e) The acceleration is zero |

Practice 4.4.4 |
---|

A car is driving around a circular track at constant speed. At one instant, the car is driving northward and sometime later the car is driving westward. What is the direction of the car’s average change in velocity during this time interval? |

(a) The car’s change in velocity will be westward. |

(b) The car’s change in velocity will be northward. |

(c) The car’s change in velocity will be westward. |

(d) The car’s change in velocity will be south of west. |

Practice 4.4.5 |
---|

A certain light truck can go around a curve having a radius of 150 m with a maximum speed of 32.0 m/s. To have the same acceleration, at what maximum speed can it go around a curve having a radius of 75.0 m? |

(a) 64 m/s |

(b) 45 m/s |

(c) 32 m/s |

(d) 23 m/s |

(e) 16 m/s |

Practice 4.4.6 |
---|

Earth is 149.6 billion meters from the Sun and takes 365 days to make one complete revolution around the Sun. Mars is 227.9 billion meters from the Sun and has an orbital period of 687 days. What is the ratio of Earth’s centripetal acceleration to Mars’s centripetal acceleration? |

(a) 1.88 |

(b) 0.809 |

(c) 1.24 |

(d) 0.430 |

(e) 2.33 |

Practice 4.4.7 |
---|

A 1000-kg car is moving at a constant speed around a circular turn with a radius of 18.5 meters. How fast must the car move to have an acceleration of 25.2 m/s^{2}? |

(a) 1.17 m/s |

(b) 466 m/s |

(c) 21.6 m/s |

(d) 34.3 m/s |

Practice 4.4.8 |
---|

A 1000-kg car is moving around a circular turn with a radius of 18.5 meters and decreasing in speed at a rate of 35.2 m/s^{2}. At the instant the car is moving at 16.8 m/s, what is the car’s total acceleration? |

(a) 38.4 m/s^{2} |

(b) 15.3 m/s^{2} |

(c) 35.2 m/s^{2} |

(d) 15.3 m/s^{2} |

(e) 38.4 m/s^{2} |

**Discuss!**

Consider how you would answer these questions. Then bring this to class for a group discussion.

A satellite has a circular orbit with an altitude of 640 km above Earth’s surface. Its period (the time for a particle to go around a closed path once) is 98.0 min. What is the satellite’s speed and radial acceleration?

A wheel 0.40 m in radius rotates at a constant rate of 180 rev/min. Find the speed and acceleration of a small stone lodged in the tread of the tire (on its outer edge).

A jet plane comes in for a downward dive as shown in the figure. The bottom part of the path is a quarter circle with a radius of curvature of 280 m. According to medical tests, pilots will lose consciousness when they pull out of a dive at an upward acceleration greater than 5.5*g*. At what speed (in m/s and mph) will the pilot black out during this dive?