Ohio Northern University
Syllabus: Integrated Math-Physics
Mike Fisher
Maria Walters

Type of Course: Combined Calculus, Physics, and Physics Laboratory course that will be team-taught

Audience: Roughly 80% freshman engineering majors
Roughly 20% math, physics, chemistry, or other majors

Goal: Become skilled in solving science/engineering problems that involve integrating one or more of the following: mathematics, physics, experimentation, and computer resources.

Brief Description of Course:
Course will span a full academic year (three quarters) and cover the following major calculus and physics courses: Differentiation (Calc 1), Integration (Calc 2), Vector Calculus (Calc 4), Newtonian Physics a (Physics 1), Thermodynamics (Physics 2), and Newtonian and Thermodynamic Physics Lab (Physics Lab 1 and 2). The course will be 9 credits per quarter, which will include a 1 credit Physics Lab each quarter. The lab will meet once weekly for 2 or 3 hours. MathCAD will be introduced and used in several laboratory sessions. The course lecture will run four days a week for two hours a day.

We plan to run one pilot course during the 00-01 academic year. Several other traditional freshman Calculus and Physics sections will be running concurrently. We are not sure if we will covert all our freshman physics and calculus courses into integrated math/physics course in the future.

UNITS

  1. Standard units and conversions
  2. Problem solving techniques & EDPIC method

Measurement/Significant figure lab

FUNCTIONS

  1. Representation of functions (Many Physics Examples)
  2. Mathematical Modeling (Modeling Physics Problems)
  3. Function Library (Many examples from Physics)
  4. GraphingCalculator/Least Squares Intro are Intro ( Will be used in Lab)
  5. MathCad (or eq.) intro (will be used in lab)

Modeling some non-linear physical phenomenon
(? Coffee filter experiment) & least square fitting

LIMITS AND RATES OF CHANGE

  1. ?Demo/motivation for understanding rates of change
  2. Limit of a Function (Many examples from Physics)
  3. Limit Laws (Examples from Physics that defy intuition)
  4. Continuity (Many examples from Physics)
  5. ?Demo/motiv. For understanding rates of change

Velocity measurement lab

DERIVATIVES/INSTANTANEOUS RATES OF CHANGE/ RECTILINEAR MOTION

  1. Slope of a tangent line problem-motivation for the definition of derivative
  2. Definition of derivative
  3. Derivatives are functions too (position to velocity to acceleration) (Can you go backwards?)
  4. Rules of derivatives (proof of product rule) (lots of word problems)
  5. Derivatives of trig functions (models of oscillations adding waves)
  6. Chain rule and implicit differentiation
  7. Free fall problems
  8. Higher order derivatives
  9. Mass on a spring (ODEs) (Newton’s 2nd law)
  10. Gravity (newton’s law)
  11. Rocket problems and other related rate problems
  12. What are differentials? (dx or delta-x)

The Graphing Lab
Measuring g

Vectors and multi-dimensions

  1. Definitions of vectors and scalars
  2. Graphical addition and subtraction
  3. 2-D coordinate systems (cartesian to polar, polar to cartesian)
  4. Vector components (free-body diagrams, force table stuff, addition, subtraction, projectile motion)
  5. 3-D coordinate systems (free body diagrams etc) (cartesian to cylindrical and spherical, and reverse)
  6. Dot and cross product

Force table lab and equilibrium

Newton’s Laws

  1. Law of inertia (gravitational mass and inertial mass)
  2. F=ma (2nd law)
  3. More ODE examples
  4. Equal but opposite (3rd law)
  5. Friction
  6. Drag (revisit coffeee filters)

Friction lab

RIEMANN INTEGRAL

  1. Spinning Bucket, deforms surface of water
  2. Definite integral
  3. Fundamental Theorem of Calculus
  4. Substitution rule
  5. Indefinite integral

Measure cross-sectional area of spinning bucket

WORK & KINETIC ENERGY

  1. Definition of work (have them do examples)
  2. Constant force problem (spring)
  3. Variable force, how to do?
  4. Force as a general mathematical function and work as a general integral
  5. Relationship between work and kinetic energy
  6. Conservation of Energy (GPE- KE)

Mass on a spring lab (damping as part 2)

Calculating Volumes

  1. Calculating volumes, motivation
  2. Volumes of revolution
  3. Volume complex objects (back to spinning bucket, solid objects)

Calculating volume of complex object (MathCAD exercise)

CONTINUOUS MEDIA

  1. Center of mass for discrete particles
  2. Extend to inhomogeneous 1-D problems
  3. Extend to 3-D COM calculations
  4. Application of 2nd law to COM

How do forces act on COM of objects

SPECIAL FUNCTIONS

  1. The Exponential (damped SHO) and derivatives
  2. Natural logarithms and derivatives
  3. Inverse trig functions
  4. Hyperbolic functions
  5. Bessel functions

MathCAD lab with special functions (revisit damping)

LINEAR MOMENTUM AND COLLISIONS (1-D)

  1. Linear momentum
  2. Conservation laws
  3. Elastic and inelastic collisions
  4. New form of 2nd law (impulse)

1-D collisions on the track

ROTATION

  1. Angular equivalents of rectilinear variables
  2. Calculate moments of inertia
  3. Techniques of integration
  4. Integration by parts
  5. Method of partial fractions/trig integrals

Centripetal acceleration

CALCULUS IN MORE THAN ONE DIMENSION

  1. Vector functions and their derivatives
  2. Parametric functions
  3. Extend mechanics to 2-D

Projectile motion

THE 0TH LAW OF THERMODYNAMICS

  1. What is temperature?
  2. Different temperature scales
  3. The meaning of Absolute Zero
  4. Thermodynamic systems and state variables
  5. Ideal Gas Law
  6. The caloric fluid

Measuring heat capacity

THE 1ST LAW OF THERMODYNAMICS

  1. Thermal Energy
  2. What is a differential? A perfect differential?
  3. The 1st law of thermodynamics
  4. Thermal expansion
  5. Heat and work
  6. Heat flow

Linear expansion problem

ENTROPY

  1. What is disorder?
  2. What is the meaning of entropy
  3. Why does time seem to have a definite direction
  4. Reversible and Irreversible Processes
  5. Dissipative processes

Finding Absolute Zero

THE 2ND LAW OF THERMODYNAMICS

  1. Entropy as a perfect differential
  2. Thermodynamic Engines
  3. Efficiency
  4. Heat Pumps

MathCAD lab on the automobile engine

MULTIVARIABLE CALCULUS

  1. Lines and surfaces in 3-D
  2. Tangents to planes
  3. Partial derivatives and functions of more than one variable
  4. Fields
  5. The gradient of a scalar
  6. The Divergence of a vector field
  7. The curl of a vector field

MathCAD lab on the del operator

FLUIDS

  1. Static fluid systems
  2. Kinetic fluid systems
  3. Ideal fluids
  4. The equations of fluid dynamics
  5. Partial differential equations

MathCAD lab on finite differencing

SIMPLE HARMONIC MOTION

  1. Waves and particles
  2. Transverse waves
  3. Longitudinal waves
  4. Trig identities
  5. Adding waves (interference)
  6. Standing waves

Melde’s experiment

GEOMETRIC OPTICS

  1. Mirrors and images
  2. Refracting surfaces
  3. Thin lenses
  4. Optical instruments

Ray tracing

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