Overview
This lecture explains the concepts of centripetal and tangential acceleration in circular motion, the derivation of centripetal acceleration, and introduces angular acceleration.
Components of Acceleration in 2D Motion
- Acceleration parallel or anti-parallel to velocity changes speed.
- Acceleration perpendicular to velocity changes direction.
Centripetal Acceleration in Circular Motion
- Centripetal acceleration is the acceleration towards the center of a circle in uniform circular motion.
- The direction of centripetal acceleration is always inward, towards the circle's center.
- The magnitude of centripetal acceleration is derived as (a_c = \frac{v^2}{r}).
- An alternative form: (a_c = r \omega^2), where (\omega) is angular velocity.
- Even at constant speed, changing direction means the object is accelerating.
- Units for centripetal acceleration: meters per second squared ((m/s^2)).
Tangential and Angular Acceleration
- Tangential acceleration changes the speed along the circular path.
- Angular acceleration ((\alpha)) is the time rate of change of angular velocity ((\omega)): (\alpha = \frac{d\omega}{dt}).
- Units for angular acceleration: radians per second squared ((rad/s^2)).
- Tangential acceleration can be expressed as (a_{tan} = r \alpha).
- The sign of angular acceleration affects whether angular velocity increases or decreases.
Example Calculations
- For an object moving at 2 m/s on a 0.5 m radius circle: (a_c = \frac{2^2}{0.5} = 8,m/s^2).
- For a car accelerating from 0 to 50 m/s in 5 s with a 100 m radius:
- Tangential acceleration: (a_{tan} = \frac{50}{5} = 10,m/s^2).
- Centripetal acceleration at 50 m/s: (a_c = \frac{50^2}{100} = 25,m/s^2).
- Total acceleration magnitude: (\sqrt{10^2 + 25^2} = 26.9,m/s^2).
Key Terms & Definitions
- Centripetal acceleration — Inward acceleration responsible for changing direction in circular motion ((a_c = v^2/r)).
- Angular velocity ((\omega)) — Rate of change of angle per unit time ((rad/s)).
- Angular acceleration ((\alpha)) — Rate of change of angular velocity ((rad/s^2)).
- Tangential acceleration — Acceleration responsible for changing the speed along the circle ((a_{tan} = r \alpha)).
Action Items / Next Steps
- Review rotational kinematics, focusing on cases with constant angular acceleration (next lecture).
- Practice calculating centripetal and tangential acceleration for various scenarios.