Circular Motion: Position, Velocity, Acceleration, and Reference Frames

Circular Motion

Angular Position, Displacement & Velocity

When a particle travels along a circular path, it is useful to describe its motion using angular quantities. These quantities help in analyzing rotational motion and relate directly to linear motion concepts.

• Angular Position (θ): The location of a particle on a circular path at a given instant, measured in radians.

• Arc Length (s): The distance traveled along the circle, related to angular position by $s = r heta$.

• Angular Displacement (Δθ): The change in angular position, given by $Δθ = θ_f - θ_i$.

• Average Angular Velocity (ωavg): The rate of change of angular displacement over time: $ω_{avg} = rac{Δθ}{Δt}$ or $ω = rac{dθ}{dt}$

• Sign Convention: $ω > 0$ for counterclockwise (ccw) rotation.

• Tangential Velocity (vt): The linear speed at the edge of the circle: $v_t = ωr$

Centripetal Acceleration

In uniform circular motion, the speed is constant but the direction of velocity changes, resulting in acceleration. This acceleration is always directed toward the center of the circle and is called centripetal acceleration.

• Centripetal Acceleration (ac): $a_c = rac{v_t^2}{r} = ω^2 r$

• Direction: Always points toward the center of the circle.

• Magnitude: Remains constant for uniform circular motion.

• Non-uniform Motion: If the speed is not constant, the acceleration may not point directly toward the center.

Instantaneous Acceleration in Circular Motion

Acceleration at a specific instant can be found by differentiating the velocity vector. For circular motion:

• Instantaneous Acceleration: $|oldsymbol{a}| = rac{dv_t}{dt}$

• Using $v_t = ωr$, and $ω = rac{dθ}{dt}$, the acceleration can be expressed as: $|oldsymbol{a}| = rac{v_t^2}{r} = ω^2 r$

Uniform Circular Motion

Uniform circular motion refers to motion along a circular path with constant angular velocity. This is a foundational model for many physical systems.

• Key Equations: $v_t = ωr$ $a_c = rac{v_t^2}{r} = ω^2 r$

• Direction: $v_t$ is tangent to the circle; $a_c$ points toward the center.

• Sign Convention: $v_t$ is positive for ccw rotation, negative for cw rotation.

• Limitation: Model fails if rotation is not steady (i.e., angular velocity is not constant).

Example: Merry-Go-Round

Consider two people riding a merry-go-round, one twice as far from the axis as the other. Their angular velocity, tangential velocity, and acceleration can be compared:

• Angular Velocity (ω): Same for both, as they complete a revolution in the same time.

• Tangential Velocity (vt): $v_t$ is proportional to radius; the person farther out has twice the tangential velocity.

• Centripetal Acceleration (ac): $a_c$ is proportional to radius; the person farther out has twice the acceleration.

Example: Acceleration of a Ferris Wheel

Given a Ferris wheel with radius $r = 9.0$ m and angular speed $ω = 2.0$ rpm, calculate the speed and acceleration of a rider:

• Convert angular speed to SI units: $ω = 2.0 ext{ rpm} = rac{2.0 imes 2 ext{π rad}}{60 ext{ s}} = 0.21 ext{ rad/s}$

• Tangential Velocity: $v_t = ωr = 0.21 ext{ rad/s} imes 9.0 ext{ m} = 1.88 ext{ m/s}$

• Centripetal Acceleration: $a_c = rac{v_t^2}{r} = rac{(1.88 ext{ m/s})^2}{9.0 ext{ m}} = 0.39 ext{ m/s}^2$

Reference Frames and Relative Motion

Reference Frames

The velocity of an object depends on the observer's frame of reference. Different observers may measure different velocities for the same object.

• Reference Frame: A coordinate system in which measurements are made.

• Relative Velocity: The velocity of object A relative to observer B is denoted $v_{A/B}$.

• Transformation of Velocity: To convert velocity from one reference frame to another, use: $v_{A/B} = v_{A/C} + v_{C/B}$

• Example: If Carlos is moving at 5 m/s on a bicycle and Amy is stationary, Amy sees Carlos moving at 5 m/s, but Carlos sees Amy as stationary.

Forces and Motion

Definition of Force

A force is a push or pull resulting from an interaction between two objects. Forces are vector quantities, meaning they have both magnitude and direction.

• Agent: The source exerting the force.

• Object: The recipient of the force.

• Contact Force: The agent and object interact by physical contact (e.g., bat hitting a baseball).

• Long-Range Force: The agent and object interact without physical contact (e.g., gravity, magnetism).

Table: Types of Forces

Type of Force	Agent-Object Interaction	Example
Contact Force	Physical contact	Bat hitting baseball
Long-Range Force	Action at a distance	Gravity acting on a mug

Direction: Pushes and pulls represent forces in opposite directions.

Vector Nature: Forces must be described by both magnitude and direction.

Additional info: The notes cover introductory concepts in rotational kinematics, reference frames, and the definition of force, suitable for a college-level physics course.