Relationships in Circular Motion Systems

This page explores relationships between linear and angular velocities in interconnected circular motion systems.

In systems with interconnected rotating parts: • Linear velocities at contact points are equal • Angular velocities of concentric systems are equal

For concentric systems: • Linear velocity increases with radius (vk > vL > vM) • Angular velocity remains constant (ωk = ωL = ωM)

Example: In a system of concentric gears, the outer gear has a higher linear velocity than the inner gear, but they share the same angular velocity.

For objects rolling without slipping: • Linear velocity of translation equals velocity of rotation at the point of contact

Highlight: The linear velocity is directly proportional to the radius in circular motion systems.

Velocity and Acceleration in Circular Motion

This page delves into velocity and acceleration calculations for circular motion.

Average velocity change in circular motion: Δv = (vfinal - vinitial) / Δt

For uniform circular motion, the magnitude of velocity remains constant, but direction changes continuously.

Formula: Centripetal acceleration: ac = v²/r = ω²r

Centripetal acceleration is always directed towards the center of the circular path.

Vocabulary: Centripetal acceleration - the acceleration of an object moving in a circular path, directed toward the center of the circle.

The page also includes examples of calculating displacement and velocity for objects in circular motion.

Highlight: In uniform circular motion, while speed remains constant, velocity is continuously changing due to the change in direction.

Circular Motion in Vertical Plane

This page discusses uniform circular motion in a vertical plane, such as in vertical loops or circular tracks.

Key points: • Speed remains constant in uniform circular motion • Net torque is zero in circular motion

For an object in vertical circular motion: T - mg = mv²/r (at the bottom) T + mg = mv²/r (at the top)

Where T is tension, m is mass, g is gravitational acceleration, v is velocity, and r is radius.

Example: For a roller coaster car in a vertical loop, the tension in the track is greatest at the bottom of the loop and least at the top.

The page also includes an example of calculating the minimum velocity required for an object to maintain circular motion at the top of a vertical loop.

Highlight: In vertical circular motion, the force of gravity affects the net force at different points in the circle, unlike in horizontal circular motion.