Topic 3-1

Topic 3-1: Circular Motion

Consider This

• Objects in two-dimensional motion can vary in both directions.

• In specific cases, coordinates can be chosen to simplify the motion to one dimension.

  • Example: Object moving on an incline can be analyzed using tilted rectangular coordinates.

• The position, velocity, and acceleration of the object will have zero y-components, focusing entirely on the x-components for kinematics.

Coordinates for Circular Motion

• A circle is defined as the set of points a fixed distance (radius R) from its center.

• When an object moves along a circular path:

  • Its distance from the center remains constant.

  • Describing this motion using polar coordinates (r, θ), where r remains constant and θ changes over time, indicates effectively one-dimensional motion in angular terms.

Angular Kinematics

• The only changing parameter is the angular coordinate θ.

• Use tools from one-dimensional kinematics without vector notation as θ is an angle:

  • Define angular position as θ(t) (in radians or degrees).

  • Angular velocity: ω(t) = dθ(t)/dt

  • Angular acceleration: α(t) = dω(t)/dt = d²θ(t)/dt²

Units

• Angular position (θ): radians (rad) or degrees (deg)

• Angular velocity (ω): rad/s or deg/s

• Angular acceleration (α): rad/s² or deg/s²

• Direction convention: counterclockwise (positive), clockwise (negative).

Relationships in Circular Motion

• Integral forms of relationships among θ, ω, and α:

  • ω(t) = ω(ti) + ∫ α(t') dt' from ti to t

  • θ(t) = θ(ti) + ∫ ω(t') dt' from ti to t

Position, Velocity, and Acceleration in Circular Motion

• Quantities used:

  • Position: r

  • Velocity: v

  • Acceleration: a

  • Angular position: θ

  • Angular velocity: ω

  • Angular acceleration: α

Translation versus Rotation

• Important differentiation:

  • Translation: Object's center-of-mass moves along a circular path (e.g., Earth around Sun).

  • Rotation: Object rotates about its center-of-mass (e.g., Earth's spin).

• Newton's 2nd Law applies differently: for translation, use linear acceleration; for rotation, a new form will be introduced in future topics.

Newton's 2nd Law for Circular Motion

• Linear acceleration, a, in the context of Newton's 2nd law, focuses on changes in linear velocity.

• Decompose linear acceleration into:

  • Centripetal Acceleration: Points toward the center, calculated as

    • a_c(t) = vt(t)²/R.

  • Tangential Acceleration: Points along the direction of motion; calculated as

    • a_t(t) = dv_t(t)/dt.

• For constant speed, only centripetal acceleration is non-zero; if speed changes, both components are present.

Relating Linear and Angular Kinematics

• Distance traveled by an object moving through an angle θ around a circle of radius R:

  • Arc length s = Rθ.

• This arc length provides the basis for defining tangential linear velocity:

  • v_t(t) = ds(t)/dt = R * dθ(t)/dt = Rω(t).

Relations of Acceleration Components:

• Centripetal: a_c(t) = Rω(t)²

• Tangential: a_t(t) = Rα(t).

Uniform Circular Motion (UCM)

• Defined as circular motion with constant speed.

• Under UCM:

  • Only centripetal acceleration acts; tangential acceleration equals zero.

  • Period T relates linear and angular velocities:

    • T = 2πR/v = 2π/ω.

• Angular frequency ω and linear frequency f related by:

  • ω = 2πf; units: rad/s (ω) and Hz (f).

Examples

Example 1: Total Linear Acceleration

• Given:

  • Radius R = 0.5 m

  • Initial angular velocity ω(t=0) = -3 rad/s

  • Angular acceleration as α(t) = α₀e^{-bt} (α₀ = 2 rad/s², b = 0.4 s⁻¹)

• Calculation of total linear acceleration at t = 10 s:

  • a_total(t) = √[a_c²(t) + a_t²(t)]

  • a_t(t) = Rα(t) and a_c(t) from angular kinematics

  • Total: a_total(10s) = 1.82 m/s².

Example 2: Speed Ratio of Satellites

• Consider two satellites with:

  • R₂ = 2R₁

• Calculate ratio of their speeds:

  • v₁ = √(GM_E/R₁)

  • v₂ = √(GM_E/R₂)

• Ratio of speeds: v₁/v₂ = √(2).