PH101_L12_Group1_

PH101 Group 1 Lecture Notes - Uniform Circular Motion

Date: 14/10/2024Instructor: Dr. Niamh Fitzgerald

1. Overview of Topics

• Uniform Circular Motion

  • Definition and principles (Sections 6.1-6.3)

• Law of Universal Gravitation

• Kepler’s Laws and Satellite Motion

2. Uniform Circular Motion

2.1 Key Concepts

• Angular Velocity (ω): Rate at which an object rotates about an axis.

• Linear Velocity (v): Speed at which an object moves along its circular path.

• Radius (r): Distance from the center of rotation to the object.

2.2 Rotation Angle

• An object traces an arc when it rotates around an axis.

• Units for Rotation Angle:

  • Radians (rad)

  • 1 complete revolution = 2π radians (360 degrees)

2.3 Relationship of Angular and Linear Velocity

• Angular Velocity Formula:

  • ω = Δθ / Δt

• Linear Velocity Formula:

  • v = Δs / Δt

  • Relationship: v = rω

• Angular velocity influences linear motion—points further from the center rotate through a greater arc length.

2.4 Example Problem: Angular Velocity of a Car Tire

• Given radius (r) = 0.300 m, speed (v) = 15 m/s.

• Calculate angular velocity (ω) and relate it to the speed at the wheel rim.

3. Centripetal Acceleration

3.1 Definition

• A change in velocity direction implies acceleration and a corresponding net force.

• Centripetal acceleration (ac) is directed towards the center of the circular path.

3.2 Formulae for Centripetal Acceleration

• Centripetal Acceleration Formula:

  • ac = v² / r

  • Alternatively: ac = rω²

3.3 Example Problem: Comparing Centripetal Acceleration to Gravity

• Determine the acceleration for a car on a curve (radius = 500 m, speed = 25 m/s) and compare it to gravitational acceleration (g).

4. Centripetal Force

4.1 Definition and Cause

• Centripetal Force: Net force acting towards the center, maintaining circular motion.

  • Formula: Fc = mac

• Can arise from tension, gravity, friction, etc.

4.2 Important Relationships

• Centripetal force can be expressed as:

  • Fc = mv² / r

  • Fc = mrω²

4.3 Example Problem: Coefficient of Friction

• Determine the coefficient of friction needed for car tires on a flat curve using Fc = f = μsN, with N = mg.

5. Banking of Curves

5.1 Concept

• On banked curves, the incline provides additional centripetal force.

5.2 Balancing Forces

• Normal force (N) has components acting towards the center (N sin θ) and balancing gravity (N cos θ).

5.3 Key Equations

• Equations used to determine the banking angle (θ):

  • N sin θ = mv² / r

  • N cos θ = mg

  • θ = tan⁻¹(v² / rg)