Class Notes on Fictitious Forces and Energy Conservation

Course Announcement

  • Next Test Details:

    • Date: November 6

    • Time: 8 AM

    • Type: Two-stage Test

    • 45 minutes for the individual test

    • 30 minutes for the group test

    • Cutoff: Content completed by the end of today

  • Time Change Reminder:

    • Clocks fall back at 2 AM on Sunday

    • Electronic devices will update automatically, unlike some physical clocks

    • Extra hour gained for preparation

Topic Overview

  • Today’s Focus: Introduction to Fictitious Forces and Energy Conservation Laws

Fictitious Forces

  • Definition: Fictitious forces are forces that people invent when they are in a non-inertial reference frame to enable the application of Newton's laws of motion.

  • Contextual Example:

    • Observing forces while on the rotating Earth

  • Inertial vs. Non-Inertial Reference Frames:

    • Newton's laws are valid only in inertial reference frames.

    • Fictitious forces come into play when working in a non-inertial frame.

Real vs. Fictitious Forces

  • Real Forces: Arise from interactions between two objects.

  • Fictitious Forces: Appear in a non-inertial reference frame and do not result from an interaction.

Example of Fictitious Forces in a Car

  • Scenario: When a car brakes, passengers feel as though they are moving forward due to the car's deceleration.

  • Analysis: An outside observer (in an inertial frame) would account for the car's acceleration effectively.

Rotating Coordinate System - Earth

  • Rotational Dynamics: The Earth rotates continuously.

    • Angular Velocity: Roughly one full rotation per day.

    • Can convert this into SI unit:

    • (2 ext{π} ext{ radians per day})

    • Radius of Earth: Approximately 6,370 km

  • Radial Acceleration:

    • Calculated using:
      A_R = R imes ext{angular velocity}^2 (points to the center of rotation)

    • Radial Distance at Kelowna (49.88° latitude):

    • R_{local} = 6370 imes ext{cos}(49.88^ ext{o})

Calculation of Radial Acceleration at Kelowna

  • Values:

    • Radius of Earth: 6.37 imes 10^6 meters

    • Cosine Value: 0.64

  • Angular Velocity (in radians):

    • ext{Total seconds in a day} = 86400 ext{ seconds}

    • ext{Angular Velocity} = rac{2 ext{π}}{86400 ext{ s}}

  • Calculated Radial Acceleration:

    • Found to be approximately 0.022 ext{ m/s}^2

Effect of Gravity and Apparent Weight

  • Gravitational Force:

    • g = rac{G imes M_E}{r^2}

    • Where G = 6.67 imes 10^{-11}, M_E = 5.9 imes 10^{24} kg

    • Validated to be approximately 9.82 ext{ m/s}^2

  • Effective Gravitational Acceleration:

    • Apparent weight differs due to radial acceleration:
      g{effective} = g - AR

Energy Conservation Laws

  • Introduction to Energy:

    • Shift from kinetic to potential energy discussed.

    • Focus on Conservation Laws: A framework to evaluate the consistency of physical systems over time.

Definitions: Conservation Laws

  • Conservation laws describe quantities that remain consistent in a closed system, essential for various physics problems.

  • Types of Conservation Laws:

    • Conservation of Momentum

    • Conservation of Energy

    • Conservation of Angular Momentum

Utilization of Conservation Laws

  • Practical Approach:

    • Enables analysis before and after events without delving into minute details between.

  • Example of Energy Conservation in Kinematics:

    • Kinematic Equation:
      v{y,f}^2 = v{y,i}^2 + 2ay

    • Rearranged to express energy forms:
      rac{1}{2}mvf^2 + mgyf = rac{1}{2}mvi^2 + mgyi

Kinetic and Potential Energy Relationship

  • Kinetic Energy (K) and Potential Energy (U) are intertwined and can be interchanged as potential energy converts into kinetic:

    • Conservation of Mechanical Energy Equation:
      K + U = ext{constant}

  • Impulse-Momentum Relationship:

    • Though complexity increases, basic principles of conservation still apply, even in circular motion/variable slopes.

    • Establish link between gravitational effects and energy conservation.

Thermal Energy Consideration

  • Inclusion of thermal energy:

    • Associated with microscopic motion of particles

    • Total energy: E = K + U + E_{thermal}

  • Work done affects energy transformations, enabling transitions between kinetic, potential, and thermal energy.

Work and Energy Transformation Principles

  • Work Done Definition: Integration of force along an object's path affects change in kinetic energy.

  • Understanding the nature of force components helps in finding work done:

  • Parallel force component relative to a path affects the ability to do work on the system.

  • Work-Force Relationship:

    • W_{ext} = ext{Work done} = ext{Force} imes ext{displacement}

    • Can only consider the parallel component of force to displacement in calculating work.

  • Work and Energy Links:

    • Work done on a system can increase or decrease its energy, following defined principles in mechanics.

  • Comparison of Energy in Closed Systems:

    • Energy transfers and transformations in isolated systems must adhere to the law of conservation of energy with changes quantifiable through work done.

Closing Remarks

  • Next Session Topics: Explore more detailed applications of conservation laws and work-energy theorems leading into discussions around practical scenarios.

  • Engagement Reminder: Acknowledgment of the potential need for refreshment after extended discussions, with clickers scheduled for activation to maintain engagement.