University:
Santa Fe College
Course:
PHY 2004 | Applied Physics 1
Academic year:

2024
Views:

482

Pages:

5
Author:

Jabin F.

Normal Coordinates: Definition: The coordinates which convert a given equation of motion (EOM) into equations of simple harmonic motion (SHM) are called normal coordinates. Property: If x is a normal coordinate, then it must satisfy the equation of SHM: x'' + ω²x = 0. Solution for Normal Coordinates: The solution for normal coordinates is given as: x = C1 cos(ωt + φ1) + C2 sin(ωt + φ2) This solution always holds for normal coordinates. Alternative Explanation: Using normal coordinates, the kinetic energy (KE) and potential energy (PE) of the system can be expressed as perfect square terms. This means that KE and PE contain no terms involving products of coordinates or products of velocities. Example: Problem: Find the frequency of normal modes of two masses connected to a spring. Solution: The kinetic energy (KE) and potential energy (PE) are expressed in terms of the displacements of the two masses (x1 and x2). The coefficient matrices for the kinetic and potential energy terms are derived. To find the frequencies of the normal modes, the equation |T - ω²V| = 0 is solved, where T is the kinetic energy matrix and V is the potential energy matrix. Calculations and Simplifications: The image continues the calculations from the previous part, solving the quadratic equation obtained from the determinant of the matrix equation: (k - m₁ω²) (k - m₂ω²) - k² = 0 This leads to two solutions for the angular frequency ω: ω = 0, ω = √(k(m₁ + m₂)/(m₁m₂)) The second solution involves the reduced mass μ, defined as: μ = (m₁m₂) / (m₁ + m₂) Finding the Angular Frequencies of Normal Modes: The image continues the calculation of the angular frequencies of the normal modes, starting from the determinant equation: |K - ω²m| = 0 Expanding the determinant and simplifying, we get: (k - ω²m)² - (k/2)² = 0 This equation can be factored as: (3k - ω²m)(k - ω²m/4) = 0 Setting each factor to zero and solving for ω, we obtain the two angular frequencies of the normal modes: ω = √(3k/m) ω = √(k/4m)