Physics 301: Module 5 - Uniform Circular Motion and Gravitation

Overview

• Focus on rotational kinematics and uniform circular motion.
• Relationship between angular displacement and the motion of particles in a circular path.

Key Concepts

Rotational Angle and Angular Velocity

• Angular Displacement (Theta, θ): Greek symbol used for angles, alternative to linear coordinates (x, y).
• Arc Length (Delta s): Length of a circular path, related to angular displacement by the equation:
  • Arc Length (s) = Radius (r) * Angular Displacement (θ)
• Conversion from linear to angular: x = r * θ
• Conversion between angles in degrees and radians: 360 degrees = 2π radians.
• Common conversions: 1 radian ≈ 57.3 degrees, rev (revolutions) = 360 degrees.

Angular Velocity (Omega, ω)

• Describes how fast an object rotates:
  • ω = Change in Angle/Change in Time
• Units: Radians per second.
• Direction: Counterclockwise (positive), Clockwise (negative).
• Relation to linear velocity: v = r * ω.*

Angular Acceleration (Alpha, α)

• Rate of change of angular velocity:
  • α = Δω/Δt
• Units: Radians per second squared.
• Similar to linear acceleration.

Tangential and Centripetal Acceleration

• Tangential Acceleration (aₜ): Related to angular acceleration:
  • aₜ = α * r
• Centripetal Acceleration (aₚ): Directed towards the center of the circular path:
  • aₚ = v²/r*

Kinematics of Rotational Motion

• Rotational variables have analogs in linear motion:
  • Displacement: θ ↔ x
  • Velocity: ω ↔ v
  • Acceleration: α ↔ a
• Key equations:
  • θ = ω_avg * t
  • ω_f = ω_i + α * t
  • θ = ω_i * t + 0.5 * α * t²
  • ω² = ω_i² + 2α * θ

Problem Solving Strategy

1. Identify rotational kinematics in the problem.
2. List knowns and unknowns; sketch if necessary.
3. Choose appropriate equations.
4. Solve for the unknowns, check units, and verify the reasonableness of answers.

Centripetal Force

• Resultant force towards the center of the circular path:
  • Fₚ = m * aₚ = m * v²/r
• Alternative forms depending on known variables:
  • Fₚ = m * r * ω²

Practical Applications

Banking of Roads

• Banking helps vehicles maintain circular paths on curves without relying solely on friction.
• Angle of banking (θ) determined by:
  • tan(θ) = v²/(r * g)*

Gravitational Force

• Newton's Law of Universal Gravitation:
  • F = G * (m₁ * m₂)/r²
• g (acceleration due to gravity) is derived from this law when considering the Earth.

Kepler's Laws

• Relates periods and radius of orbit for objects orbiting massive bodies.

Closing Remarks

• Review concepts and prepare for problem-solving and upcoming test.