Study guide for exam 1 (midterm exam)

Exam 1 (midterm) study guide

You may prepare one page of notes to bring to the exam, and you may consult your textbook. Bring writing instruments and a calculator.

Vectors, dot products, cross products, addition and subtraction. Separating vector equations into components.
Newton's laws, and what constitutes an "inertial reference frame".
Using Newton's second law, $\myv F=m\ddot{\myv r}$, in Cartesian and non-cartesian coordinate systems.

Some practice problems in Chapter 1: 2, 4, 6, 24, 25, 30, 35, 44

Setting up equations of motion (differential equation) with a velocity-dependent force (air resistance or Lorentz force).
Solving differential equations by using separation of variables.
Qualitative characteristics of motion with linear or quadratic resistance.
Setting up equations of motion with electric and magnetic fields.
Qualitative behavior of a charged particle moving in an electric field; moving in a magnetic field.
Origin of linear and quadratic resistance. Dependence on what characteristics of the body / medium.
Coupled differential equations.
Complex numbers as 2-d vectors in Cartesian ($a+bi$) and polar ($Re^{i\theta}$) representations.
Use of Complex variables to represent circular motion and periodic motion (also in Chapter 5) .

Some practice problems in Chapter 2; 5, 7, 8, 15, 23, 46

Conservation of linear and angular momentum; Solving problems using conservation of momentum.
Expressions for total linear / angular momentum of a system of particles.
Calculation and use of: Center of mass; Moment of inertia; External and internal forces; Torques.
Using the rocket equation.

Some practice problems in Chapter 3: 2, 7, 15, 16, 29, 31, example 3.3.

Kinetic energy and the "Work-Kinetic Energy" theorem.
Calculating work done by a force from the line (path) integral $W(1\to 2)= \int_1^2\myv F\cdot d\myv r$.
Conservative forces: Characteristics of a conservative force; How to detect if a force field is conservative, the curl test: $\myv{\grad}\times \myv F\stackrel{?}{=} 0$; Connection to potential energy, $U$.
For conservative forces, using $\myv F=-\myv\grad U$; Finding $U$ from $\myv F(\myv r)$ (either guessing and checking, or integrating...); The vast simplification of work integrals for conservative forces: $\Delta W=\int_{\myv r_1}^{\myv r_2}\myv F\cdot d\myv r=U(\myv r_1)-U(\myv r_2)$.
Mechanical energy, $E=T+U$; Using conservation of energy, $\dot E=0$ to find an equation of motion.
Analyzing a physical system to come up with an expression for potential energy, kinetic energy.
For linear and curvilinear 1-d systems, analyzing motion from a graph of $U(x)$: equilibrium points, detecting stable/unstable equilibria, turning points, max- and minimum speeds, (and eventually relative frequencies of oscillation for small amplitude motions).
Central forces
Using gradient / cross product definitions in Cartesian and non-Cartesian coordinate systems (particularly cylindrical and spherical-polar coordinates.

Some practice problems in Chapter 4: 2, 10, 12, 13, 15, 20, 23, 34, 36

The exam will cover sections 5.1, 5.2, and 5.4.

Taylor expansion of any continuous potential energy function near an equilibrium is $$U(x)\approx U(x_0) + \frac 12k(x-x_0)^2$$ where the equilibrium position, $x_0$, of the potential energy function satisfies $\left.\frac{\del U}{\del x}\right|^{x=x_0}=0$, and $$k\equiv\left.\frac{\del^2 U}{\del x^2}\right|^{x=x_0}$$
The harmonic oscillator differential equation $$m\ddot x(t)=-kx(t).$$ or $$\ddot x = -\frac km x=-\omega_0^2x.$$ Different forms of the solutions include sine + cosine function, a sine function with a phase shift, the real part of a complex exponential. In each case there are two constants in the general solution which can be fixed by specifying initial conditions (typically the position and velocity at $t=0$, but any two constraints can work).
Maximum kinetic energy, maximum potential energy of a simple harmonic oscillator.
Analyzing physical systems to find a restoring force.
Damped harmonic oscillation satifying the differential equation $$\ddot x +2\beta\dot x+\omega_0^2x=0.$$ The general solutions for weak damping, critical damping, and strong damping, and how the relation between $\beta$ and $\omega_0$ determines which of these regimes will satisfy the differential equation.

Some practice problems in chapter 5: 3, 5, 6, 22. Examples 5.1 and 5.2.