Lecture 7

Circular Motion and Gravitation

Introduction

• Presenting the concepts of circular motion and gravitation.

• Dr. Val G. Rousseau – Xavier University of Louisiana.

Angular Displacement, Velocity, and Acceleration

Key Quantities

• Angular displacement, angular velocity, and angular acceleration are crucial for understanding circular motion.

Arc Length

• Arc Length (s):

  • Distance measured along the circumference of a circle between two points (A and B).

Radians

• Measurement of angles:

  • Radians provide an alternative to degrees.

  • Definition: A radian is the angle θ formed by an arc length (s) equal to the radius (R).

  • Relationship: Full circle (360°) corresponds to 2π radians.

  • Formula:

    • p = 2πR

    • For a complete rotation:

      • 2π radians = 360°.

Angular Displacement

Definition

• Angular displacement (Δθ):

  • Defined as the change in rotation angle.

  • Formula:

    • Δθ = θf - θi

  • Directionality:

    • Positive if counterclockwise.

    • Negative if clockwise.

Angular Velocity

Definition

• Angular velocity (ω):

  • Expresses how fast the angular displacement changes with time.

  • Average Angular Velocity:

    • ω_avg = Δθ/Δt

  • Instantaneous Angular Velocity:

    • Related to linear speed (v), with v = Rω.

Angular Acceleration

Definition

• Angular acceleration (α):

  • Describes the rate at which the angular velocity changes over time.

  • Angular Acceleration Equation:

    • α = Δω/Δt

Rotational Kinematics

Relationships

• The angular quantities (Δθ, ω, α) follow the same equations and relationships as linear quantities (Δx, v, a) under constant angular acceleration conditions.

Tangent Quantities

Definition

• Tangent references instantaneous velocity tangential to the trajectory in circular motion.

• Tangent Line:

  • Touches the circle at a single point and is perpendicular to the radius.

Examples of Circular Motion

Example 1: Vinyl Record Turntable

• Case Reference:

  • Angular Velocity: Given as 33 ⅓ rpm.

  • Conversions: Required to find angular velocity in radians per second and the linear speed at a distance from the rotation axis.

Example 2: Bicycle Wheels

• Case Reference:

  • Slow down from 9.20 m/s to rest over 85.0 m.

  • Calculations include finding angular velocity and angular acceleration, involving direct applications of kinematic equations.

Centripetal Force

Definition

• Centripetal force:

  • A net force causing centripetal acceleration toward the center during circular motion.

The Universal Law of Gravitation

Definition

• Newton's Law:

  • Every particle attracts every other particle with a force along the line joining them, directly proportional to their masses and inversely proportional to the square of the distance between them.

  • Gravitational force (F):

    • F = G (m1*m2/r²)

  • G: Universal Gravitational Constant.

Conclusion

• Understanding these fundamental concepts of angular motion and gravitation is essential for solving problems in physics related to circular contexts.