Chapter 2: Forces - Centripetal Motion & Universal Gravitation

Centripetal Motion

• Definition: Traveling in a circular path. Even a fraction of a circle (e.g., a car taking a turn or a banked curve on a road) involves centripetal motion.

• Vocabulary: The standard terms are still used: displacement, velocity, acceleration.

• Speed in Centripetal Motion: Typically constant, as seen with recommended speed limits for turns on roads, which are calculated based on centripetal motion principles.

• Etymology: The term "centripetal" means "center seeking."

• Car Example (Constant Speed, Changing Velocity):

  • Imagine a car in a turn with cruise control set; the speed (magnitude of velocity) is constant.

  • However, velocity is a vector quantity, meaning it has both magnitude and direction.

  • At any instant, the car's velocity vector is tangent to its circular path.

  • Even if the speed remains the same, the direction of the velocity vector is constantly changing as the car moves around the curve.

• Acceleration: Changing direction constitutes acceleration.

  • The steering wheel, not just the gas pedal, causes acceleration because it changes the direction of motion.

  • Acceleration is defined as the difference in velocity over time ($a = \frac{\Delta v}{\Delta t}$), and this includes changes in direction.

  • In centripetal motion, the acceleration ($a_c$) is always directed towards the center of the circle.

• Centripetal Force ($F_c$):

  • According to Newton's Second Law, an acceleration is caused by a net force ($F = ma$).

  • Therefore, a centripetal force is required to cause the centripetal acceleration and keep an object in circular motion.

  • The centripetal force ($F_c$) also points towards the center of the circle.

  • Without this force, the object cannot stay in a circular path (it would fly off tangent to the circle).

  • Examples:

    • Cars on the Road: The friction between the tires and the road provides the necessary centripetal force for the car to turn.

Centripetal vs. Centrifugal Force

• Mass on a String Analogy:

  • Imagine twirling a mass (e.g., a $500 ext{ g}$ lab mass) on a string above your head. The length of the string is the radius ($r$).

  • You feel a constant pull or