Static Equilibrium and Problem-Solving

Static Equilibrium

• Static equilibrium occurs when a rigid body experiences zero resultant external force and zero resultant external torque about any axis.
• Mathematically, the conditions are represented as ΣF = 0 (resultant external force) and Στ = 0 (resultant external torque).

Problem-Solving Strategy for Static Equilibrium

• Choose the system to be analyzed.
• Draw a free-body diagram showing all external forces acting on the system.
• Select a coordinate system.
• Apply equilibrium conditions: ΣFx = 0, ΣFy = 0, and Στ = 0 about a chosen axis.
• Solve the resulting system of equations for the unknown quantities.
• Check that the answers are reasonable and consistent with the information.

Uniform Beam Example

• A beam of length L and mass m, supported by a hinge ($P$) and cable ($Q$), has a block of mass M suspended from its end.
• To find the tension $T$ in the cable and the reaction force components $R_x$ and $R_y$ at the hinge, apply the following equilibrium conditions:
  • ΣFx = Rx - T cos θ = 0
  • ΣFy = Ry + T sin θ - mg - Mg = 0
  • Στ = T sin θ (3L/4) - Mg(L) - mg(L/2) = 0 (torques about point P)
• By solving the equations, the tension T, and reaction components $R_x$ and $R_y$ are:
  • T = 4(M + m/2)g / (3 sin θ)
  • Rx = 4(M + m/2)g / (3 tan θ)
  • Ry = (m - M)g / 3

Key Notes on Equilibrium

• The choice of axis for torque calculation is arbitrary, and should be chosen to simplify calculations.
• Ensure all relevant forces are included in the free-body diagram.
• Pay attention to the sign of the torques.
• Units in the equations must be consistent.

How To Succeed in Chemistry

• Be prepared for each class.
• Read the textbook.
• Complete assigned homework.
• Attend class.
• Ask questions when unsure.
• Get help immediately when struggling.
• Keep up with the material/content as it is provided.
• Always check your work.
• Think critically.
• Memorize key information and concepts.

Chemistry Prerequisite Knowledge

• Basic Mathematics and Algebra skills are required.
• Knowledge of significant figures is critical.
• Understanding of dimensional analysis is important.
• Nomenclature is a requirement.

Matter

• Matter is anything that has mass and takes up space.
• The states of matter are solid, liquid, and gas.
• Matter consists of elements, compounds, and mixtures.

Dimensional Analysis

• Units are important in calculations.
• Always include units in all calculations.
• To convert 8.00 ft to inches: 8.00 ft * (12 inches / 1 ft) = 96.0 inches.

Significant Figures Rules (Sig Figs)

• All non-zero digits are significant (e.g., 1.56 has 3 sig figs).
• Leading zeros are not significant (e.g., 0.0025 has 2 sig figs).
• Trailing zeros are significant only if the number contains a decimal point (e.g.,100 has 1 sig fig, 100. has 3 sig figs, 100.00 has 5 sig figs).
• Zeros between non-zero digits are always significant (e.g., 1.05 has 3 sig figs).
• Exact numbers have an unlimited number of significant figures (e.g., 1 inch = 2.54 cm).

Significant Figures in Calculations

• In multiplication and division, the answer cannot have more significant figures than the original numbers with the fewest significant figures (e.g., 1.36 * 4.2 = 5.712 ≈ 5.7 with 2 sig figs).
• In addition and subtraction, the answer cannot have more digits to the right of the decimal point than the original number with the fewest digits to the right of the decimal point (e.g., 1.025 + 2.1 = 3.125 ≈ 3.1).

Atoms

• Atoms consist of protons, neutrons, and electrons.
• Protons: positive charge, mass = 1 amu, located in the nucleus.
• Neutrons: neutral charge, mass = 1 amu, located in the nucleus.
• Electrons: negative charge, mass = 0 amu, located in orbitals (electron cloud).

Key Atomic Numbers

• Atomic number (Z) is the number of protons in the nucleus of an atom.
• Mass number (A) is the sum of the number of protons and neutrons in the nucleus.
• Isotopes are atoms with the same number of protons but different numbers of neutrons.

Ions

• Ions are charged atoms.
• Cations are positively charged.
• Anions are negatively charged.

Periodic Table

• Groups are the vertical columns.
• Periods are the horizontal rows.
• Metals are located on the left side and are good conductors; loses electrons to form cations.
• Nonmetals are on the right side;poor conductors; gains electrons to form anions.
• Metalloids are along the staircase line and have properties of both metals and nonmetals.

Ionic Compounds

• Compounds are formed between a metal and a nonmetal.
• Name the metal first, then the nonmetal with the suffix -ide (e.g., NaCl = Sodium chloride, $Al_2O_3$ = Aluminum oxide.)
• Elements that form more than one type of ion have Roman numerals to indicate the charge (e.g., $FeCl_2$ = Iron(II) chloride, $FeCl_3$ = Iron(III) chloride).
• Polyatomic ions should be memorized.

Molecular Compounds

• Molecular structure is nonmetal bonded to a nonmetal.
• Prefixes indicate the number of each element (e.g., $N_2O_4$ = Dinitrogen tetroxide, $PCl_5$ = Phosphorus pentachloride).

Prefixes and Numbers

• Mono- = 1
• Di- = 2
• Tri- = 3
• Tetra- = 4
• Penta- = 5
• Hexa- = 6
• Hepta- = 7
• Octa- = 8
• Nona- = 9
• Deca- = 10

Acids

• Acids produce $H^+$ ions when dissolved in water.
• Binary acids use the prefix hydro- followed by the nonmetal name with the suffix -ic (e.g., HCl = hydrochloric acid).
• Oxyacids: if the polyatomic ion ends in -ate, change the ending to -ic acid (e.g., $HNO_3$ = nitric acid).
• If the polyatomic ion ends in -ite, change the ending to -ous acid (e.g., $HNO_2$ = nitrous acid).

Chemical Reactions

• To balance chemical equations: Write the unbalanced equations, balance the equation by adding coefficients, and check that the equation is balanced.

Stoichiometry

• Relationship between reactants and products is quantitative.
• Mole is the amount of substance containing $6.022 * 10^{23}$ representative particles.
• Molar mass is the mass of one mole.
• Coefficients determine the mole ratio between reactants and products.

Limiting Reactant

• The reactant that is completely consumed and limits the reaction.
• Theoretical yield is the product amount from complete conversion of the limiting reactant.
• Percent yield is (Actual Yield / Theoretical Yield) * 100%.