Overview
This lecture explains the conditions for circular motion, the concept and calculation of centripetal force, and applies these ideas to real-world examples such as satellites, vehicles, and amusement park rides.
Conditions for Circular Motion
- Circular motion requires a force that acts at 90° to the object's velocity, always pointing toward the center of the circle.
- The force causes acceleration, which changes only the direction (not speed) of velocity in uniform circular motion.
- This force is called the centripetal force.
Centripetal and “Centrifugal” Forces
- Centripetal force: always directed towards the center of the circle, responsible for circular motion.
- “Centrifugal force” is a perceived, not real, outward force due to inertia; only centripetal force acts physically.
Circular Motion Examples
- For satellites, gravity provides the centripetal force pulling them toward the Earth's center.
- For cars on roundabouts, friction between tires and road supplies centripetal force.
- In amusement rides, tension or structural forces act as the centripetal force.
- If centripetal force is suddenly removed (e.g., string snaps), the object moves off tangentially.
Key Equations for Circular Motion
- Acceleration in a circle: ( a = v^2 / r )
- Centripetal force: ( F = m v^2 / r )
- Speed of object in circular motion: ( v = 2\pi r / T ) (where ( T ) is the time period)
- Frequency ( f = 1 / T ); speed using frequency: ( v = 2\pi f r )
- Angular velocity: ( \omega = 2\pi f = 2\pi / T ) (radians per second)
- Alternative centripetal force equation: ( F = m \omega^2 r )
Applications: Satellites, Planes, and Vertical Loops
- Geostationary satellites: time period ( T = 24 ) hours, radius measured from Earth's center.
- Use ( F = m \omega^2 r ) to find centripetal force without knowing linear speed.
- For banked turns (e.g., airplanes, cars): resolve forces into vertical and horizontal components.
- Horizontal component provides centripetal force: ( m v^2 / r ).
- ( v = \sqrt{g r / \tan\theta} ) for banking angle ( \theta ).
- For vertical loops (rollercoasters):
- At top: centripetal force needed is ( m v^2 / r ), weight helps reduce required support force.
- At bottom: support force must overcome both centripetal need and weight.
Key Terms & Definitions
- Centripetal Force — force always directed toward the center of the circle, causing circular motion.
- Centrifugal Force — apparent outward force felt in rotating frames, not a real force.
- Angular Velocity (( \omega )) — rate of change of angle in radians per second.
- Time Period (( T )) — time taken for one complete revolution.
- Frequency (( f )) — number of revolutions per second.
Action Items / Next Steps
- Practice rearranging the centripetal force equation for variables like speed and radius.
- Review the application of forces in banked turns and vertical loops.
- Ensure understanding of how to apply equations using angular velocity, frequency, and time period.