Overview
This lecture explains the kinematics, dynamics, key formulas, concepts related to uniform & non-uniform circular motion, and practice problems of Circular Motion.
Introduction to Circular Motion
- Circular motion — an object moves along a circle with a fixed radius.
- Circular motion involves kinematics (velocity, acceleration, displacement) and dynamics (force).
- Circular motion can occur in horizontal or vertical planes.
Linear vs Angular Motion
- Linear motion involves displacement (s), velocity (v), acceleration (a).
- Angular motion involves angular displacement (θ), angular velocity (ω), angular acceleration (α).
- Linear - s, u, v, a; Angular - θ, ω, α.
Important Formulas & Relations
- Linear velocity: ( v = \frac{\Delta x}{\Delta t} )
- Angular velocity: ( \omega = \frac{\Delta \theta}{\Delta t} )
- ( v = r \omega ) (relation between linear & angular velocity)
- Acceleration (Uniform Circular Motion): Centripetal acceleration ( a_c = \frac{v^2}{r} = r \omega^2 )
- Equations of Motion (linear ↔ angular): ( s \leftrightarrow \theta,, u \leftrightarrow \omega_i,, a \leftrightarrow \alpha )
Speed, Velocity, Acceleration in Circular Motion
- In uniform circular motion speed is constant; magnitude of velocity is constant, direction changes.
- In non-uniform circular motion both speed and velocity change.
- Acceleration is divided into centripetal (radial) & tangential components.
- Total acceleration: ( a = \sqrt{a_c^2 + a_t^2} )
Centripetal Force & Applications
- Centripetal force is essential in circular motion.
- ( F_{centripetal} = m \frac{v^2}{r} = m r \omega^2 )
- Various existing forces (spring, gravitational, electrostatic) act as centripetal force in different cases.
- Spring force: ( F = kx ) (x = elongation)
Practice Questions & Applications
- Finding angular velocity, tension for mass attached to string in horizontal/vertical circle.
- Method of using centripetal force in problems.
- Difference between average velocity and instantaneous velocity.
Key Terms & Definitions
- Circular Motion — motion of an object along a circle.
- Linear Displacement (s) — direct changed distance.
- Angular Displacement (θ) — angle turned (radians).
- Linear Velocity (v) — speed changing in a straight line.
- Angular Velocity (ω) — angle turned per unit time.
- Acceleration (a, α) — rate of change of speed (linear, angular).
- Centripetal Force — force acting towards the center.
Action Items / Next Steps
- Learn vertical circular motion and more problems in the next class.
- Revise previous concepts (linear/angular motion equations, force relations).
- Homework: Practice given examples to find tension, centripetal force in calculations.