(455) Centripetal acceleration and force [IB Physics SL/HL]

Centripetal Acceleration and Force

Introduction

• Centripetal means center-seeking.

• Discusses the concept of circular motion and its implications in physics.

Acceleration

• Definition: Acceleration is the change in velocity over time and is a vector (has magnitude and direction).

• Acceleration can occur from changes in both speed and direction.

• Example: A ball rolling in a straight line accelerates if its direction changes, even if its speed remains constant.

Concept Visualization

• Visualize a ball on a flat surface.

• If pushed in a straight line, it continues straight; tapping it laterally causes it to change direction towards the center, illustrating centripetal force.

• Centripetal Force: Always directed toward the center of the circular path.

Centripetal Acceleration

• Formula: ( a_c = \frac{v^2}{r} ) where:

  • ( a_c ): Centripetal acceleration (m/s²)

  • ( v ): Linear speed (m/s)

  • ( r ): Radius of the circle (m)

• Both speed and radius must be considered for motion in a circle, reaffirming that direction change results in acceleration.

Centripetal Force

• Derived from Newton's second law: ( F = ma )

• Centripetal force formula: ( F = m \frac{v^2}{r} )

  • Where F is in Newtons, mass in kg, speed in m/s, and radius in meters.

  • Provides insight into the force required to maintain circular motion.

Angular & Linear Speed Relations

• ( v = \omega r ) where ( \omega ): angular speed in radians/second.

• Equivalent force formula: ( F = m \omega^2 r )

Example: Earth's Centripetal Acceleration

• Radius of Earth: ( R = 6.4 \times 10^6 , m )

• Calculate centripetal acceleration experienced due to Earth's rotation:

  1. Use ( A = \frac{v^2}{r} )

  2. Determine linear speed ( v ): Circumference of circle divided by the period of one rotation (24 hours).

  3. Convert 24 hours to seconds: 24 hours = 86,400 seconds.

  4. Substitute values to find acceleration: ( A = 4 \pi^2 \frac{R}{T^2} )

  5. Calculation yields: ( A \approx 0.0338 \text{ m/s²} )

Conclusion

• The resulting acceleration is significantly less than standard gravitational acceleration, indicating a very small centripetal acceleration despite the Earth's rotation.