Overview
This lecture introduces and explains centripetal acceleration and centripetal force in circular motion, including their directions, real-life examples, and key equations.
Circular Motion and Velocity
- In circular motion, the velocity vector is always tangent to the circle.
- Although the speed can remain constant, the direction of the velocity vector changes continuously.
- A change in direction of velocity means the object is accelerating.
Centripetal Acceleration
- Acceleration in circular motion always points toward the center of the circle (centripetal acceleration).
- Centripetal acceleration is caused by the continuous change in direction of the velocity vector.
- The formula for centripetal acceleration is ( a_c = \frac{v^2}{r} ) or ( a_c = r\omega^2 ), where ( v ) is tangential velocity, ( r ) is radius, and ( \omega ) is angular velocity.
Centripetal Force
- According to Newton's second law, a net force is required for acceleration.
- Centripetal force is the resultant force that causes an object to follow a circular path.
- The formula for centripetal force is ( F_c = m\frac{v^2}{r} ) or ( F_c = mr\omega^2 ), where ( m ) is mass.
- Centripetal force always points toward the center of the circle and is perpendicular to the object's velocity.
Sources of Centripetal Force (Examples)
- Tension in a string provides centripetal force for a ball swung in a circle.
- Friction between tires and the road provides centripetal force for a car turning.
- Gravity supplies centripetal force for planets orbiting the Sun.
- Normal force and gravity can both act as centripetal forces (e.g., roller coaster loops).
Key Terms & Definitions
- Centripetal acceleration ((a_c)) — Acceleration directed toward the center of a circular path.
- Centripetal force ((F_c)) — Net force required to keep an object moving in a circle, directed toward the center.
- Tangential velocity ((v)) — Velocity at a tangent to the circle at a given point.
- Radius ((r)) — Distance from the center of the circle to the object.
- Angular velocity ((\omega)) — Rate of change of angle, measured in radians per second.
Action Items / Next Steps
- Memorize and practice using the formulas for centripetal acceleration and force.
- Identify the source of centripetal force in different circular motion scenarios.
- Prepare for upcoming examples involving calculations with force, velocity, and radius.
- Optional: Watch the bonus video for derivation of the centripetal acceleration formula.