P191 HOMOGENOUS

Homogeneous Equations

Definition

• For any equation to be valid, it must be homogeneous.

• A homogeneous equation has equal base units on both the left and right sides.

Example Equations

1. S = ut + ½ at²

   • Variables:

     • S: displacement

     • u: initial velocity

     • a: acceleration

     • t: time

   • Base Units Evaluation:

     • Left Side: S = m (meter)

     • Right Side:

       • ut: m/s * s = m

       • ½ at²: m/s² * s² = m

   • Conclusion:

     • m = m + m, therefore confirmed homogeneous.

Example Problems

Problem A

• Equation:

  • v² = u² + 2as

• Task: Show that the equation is homogeneous by examining base units.

Problem B

• Equation:

  • T = 2π √(I / (Mg h))

• Task: Determine the base units of I, where g = acceleration due to gravity, and h = length.

Problem C

• Equation:

  • P = ½ p <c²>

• Task: Use base units to confirm homogeneity. c² represents speed squared.

Problem D

• Equation:

  • v = ab + bt + cd + t

• Task: Find S.I. units and physical quantities of a, b, c, and d.

Dimensional Analysis

Basic Dimensions

1. Mass

   • Unit: Kilogram (kg)

   • Symbol: M

2. Electric Current

   • Unit: Ampere (A)

   • Symbol: I

3. Length

   • Unit: Meter (m)

   • Symbol: L

4. Time

   • Unit: Second (s)

   • Symbol: T

5. Temperature

   • Unit: Kelvin (K)

   • Symbol: K

Additional Examples

Example 2

1. Equation:

   • E = gh² + 3st + K

   • Task: Determine the dimensional units of K.

2. Equation:

   • F = ax + bt² + C

   • Task: Find dimensions of ax and bt².

Example 3

1. Pressure Relation:

   • Relate P to v, a, and E using dimensional analysis.

2. Centripetal Force Relation:

   • Examine the relationship for F, m, v, and r.