Title 98 Simple Harmonic Motion.pdf ✅

MSTC 98: Simple Harmonic Motion 1. Definition of Terms • Periodic Motion – This type of motion repeats itself after an equal interval. • Oscillation Motion- It is a type of periodic motion bounded between two extreme points. • Simple Harmonic Motion – A special case of oscillation, along with a straight line between the two extreme points. 2. Equations in Simple Harmonic Motion 2.1. Hooke’s Law This is the fundamental equation of simple harmonic motion. It expresses as F = −kx Here, F is the restoring elastic force, k is the spring constant, and x is the displacement. 2.2. Differential Equation From Newton’s second law of motion, F = ma, then the differential equation for simple harmonic motion is d 2x dt 2 = − k m t This results in the solution, the object's position at any point in time. x(t) = x0 cos(√ k m t) + v0√ m k sin (√ k m t) Or x(t) = A cos(ωt − φ) Where A = √c1 2 + c2 2 , tanφ = c2 c1 , and ω = √ k m 2.3. Velocity function 2.3.1. At any point From the differentiation of the position function, v(t) = dx dt = −Aω sin(ωt − φ) 2.3.2. Maximum speed At the equilibrium point, the maximum speed is a v = ωA.