Algebra for physics Knowledge-Based Study

Study notes

1. Rearranging Algebraic Equations

   In physics, relationships between physical quantities are expressed using symbolic equations rather than isolated numbers. A fundamental skill is isolating a target variable symbolically before substituting numerical values with their units.

   Core Rules for Rearranging Equations

   • Addition and Subtraction: To move a term across the equals sign, apply the inverse operation to both sides of the equation. For example, in the kinematic relation $v = v_0 + at$, subtracting $v_0$ from both sides yields $v - v_0 = at$.
   • Multiplication and Division: To isolate a factor, divide or multiply both sides by the unwanted factors. From $v - v_0 = at$, dividing both sides by $a$ (assuming $a \neq 0$) gives the expression for time: $$t = \frac{v - v_0}{a}$$
   • Fractional Expressions: When a variable appears in a denominator, multiply both sides by the denominator first to eliminate fractions. For density $\rho = \frac{m}{V}$, multiplying by $V$ gives $m = \rho V$, which can then be solved for volume as $V = \frac{m}{\rho}$.

   Equations with Exponents and Common Factors

   Many physical laws involve powers or require factoring out common quantities before isolation.

   Working with Roots and Powers

   When the target variable is raised to an exponent, isolate the power term completely before taking the corresponding root.

   • Consider kinetic energy: $K = \frac{1}{2}mv^2$.
   • Multiply both sides by 2: $2K = mv^2$.
   • Divide both sides by mass $m$: $\frac{2K}{m} = v^2$.
   • Take the principal square root: $$v = \sqrt{\frac{2K}{m}}$$

   Factoring Out Common Terms

   If the unknown variable appears in multiple terms, group those terms on one side and factor the variable out.

   For example, in heat transfer equilibrium: $Q = mc(T_f - T_i) = mcT_f - mcT_i$. When solving for final temperature $T_f$ in combined systems such as $m_1 c_1 (T_f - T_1) + m_2 c_2 (T_f - T_2) = 0$:

   1. Expand the terms: $m_1 c_1 T_f - m_1 c_1 T_1 + m_2 c_2 T_f - m_2 c_2 T_2 = 0$.
   2. Group terms containing $T_f$: $T_f (m_1 c_1 + m_2 c_2) = m_1 c_1 T_1 + m_2 c_2 T_2$.
   3. Divide by the grouped coefficient: $$T_f = \frac{m_1 c_1 T_1 + m_2 c_2 T_2}{m_1 c_1 + m_2 c_2}$$

   Direct and Inverse Proportions

   Algebra in physics allows us to predict how changing one quantity impacts another without performing full numerical calculations.

   Types of Proportional Relationships

   • Direct Proportionality ($y \propto x$): If $y = kx$, doubling $x$ causes $y$ to double. For instance, Newton's second law $F = ma$ shows that net force is directly proportional to acceleration for a constant mass.
   • Inverse Proportionality ($y \propto \frac{1}{x}$): If $y = \frac{k}{x}$, doubling $x$ reduces $y$ to half its original value. Acceleration $a = \frac{F}{m}$ is inversely proportional to mass under a constant net force.
   • Inverse-Square Proportionality ($y \propto \frac{1}{r^2}$): In Newton's law of universal gravitation, $F_g = G\frac{m_1 m_2}{r^2}$. If the separation distance $r$ is tripled, the gravitational force decreases by a factor of $3^2 = 9$, becoming $\frac{1}{9}$ of its initial magnitude.

   Solving Simultaneous Equations

   Physical situations often involve multiple interacting objects or constraints, leading to a system of equations with two or more unknowns.

   Method of Substitution

   A standard approach is isolating one unknown in the first equation and substituting it into the second.

   Consider two masses connected by a light string over a frictionless pulley (Atwood machine), yielding the equations:

   1. $T - m_1 g = m_1 a$
   2. $m_2 g - T = m_2 a$

   To determine the acceleration $a$ algebraically without calculating tension $T$ first:

   • Isolate $T$ from equation (1): $T = m_1 g + m_1 a$.
   • Substitute this expression into equation (2): $m_2 g - (m_1 g + m_1 a) = m_2 a$.
   • Collect all terms involving $a$ on one side: $m_2 g - m_1 g = (m_1 + m_2)a$.
   • Solve for $a$: $$a = \frac{(m_2 - m_1)g}{m_1 + m_2}$$